Handbook of Optical Fibers [1st ed. 2019] 9789811070853, 9789811070877

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Gang-Ding Peng Editor

Handbook of Optical Fibers

Handbook of Optical Fibers

Gang-Ding Peng Editor

Handbook of Optical Fibers With 1544 Figures and 89 Tables

123

Editor Gang-Ding Peng Photonics and Optical Communications School of Electrical Engineering and Telecommunications University of New South Wales Sydney, NSW, Australia

ISBN 978-981-10-7085-3 ISBN 978-981-10-7087-7 (eBook) ISBN 978-981-10-7086-0 (print and electronic bundle) https://doi.org/10.1007/978-981-10-7087-7 © Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

This research- and application-oriented book covers main topical areas of optical fibers. The selection of the chapters is weighted on technological and applicationspecific topics, very much a reflection of where research is heading to and what researchers are looking for. Chapters are arranged in a user-friendly format essentially self-contained and with extensive cross-references. They are organized in the following sections: Optical Fiber Communication Solitons and Nonlinear Waves in Optical Fibers Optical Fiber Fabrication Active Optical Fibers Special Optical Fibers Optical Fiber Measurement Optical Fiber Devices Optical Fiber Device Measurement Distributed Optical Fiber Sensing Optical Fiber Sensors for Industrial Applications Polymer Optical Fiber Sensing Photonic Crystal Fiber Sensing Optical Fiber Microfluidic Sensors Several sections and chapters of the book show how diverse optical fiber technologies are becoming now. We envisage that many new optical fibers under development will find important future applications in telecommunications, sensing, and so on. Though we have been trying to cover most relevant and important topics in optical fibers, some topics may have not been presented. All the authors are either pioneers or leading researchers in their respective areas. Their chapters have reflected well the excellent research work, technology deployment, and commercial application of their own and others. Hence this handbook, as a new entry to the Springer Nature’s Major Reference Works (MRWs), will be useful for researchers, academics, engineers, and students to access expertly summarized specific topics on optical fibers for research, education, and learning purposes. v

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Preface

There could be technical and grammatical errors in this book. Please feel free to send your correction, advice, and feedback to us. One key feature of the Springer Nature’s MRWs is to ensure continuing update and improvements. I would take this opportunity to express by deepest gratitude to all my colleagues, either as section editors or authors, for their hard work and great contribution to this book. I also would like to thank the Springer Nature editors and staff, especially Dr. Stephen Siu Wai Yeung and Dr. Juby George, for their kind and professional support throughout this book project. July 2019

Gang-Ding Peng

Contents

Volume 1 Part I Optical Fiber for Communication . . . . . . . . . . . . . . . . . . . . . . . . . 1

Single-Mode Fibers for High Speed and Long-Haul Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . John D. Downie, Ming-Jun Li, and Sergejs Makovejs

1

3

2

Multimode Fibers for Data Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . Xin Chen, Scott R. Bickham, John S. Abbott, J. Doug Coleman, and Ming-Jun Li

41

3

Multi-core Fibers for Space Division Multiplexing . . . . . . . . . . . . . . Tetsuya Hayashi

99

4

Optical Coherent Detection and Digital Signal Processing of Channel Impairments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ezra Ip

Part II

147

Solitons and Nonlinear Waves in Optical Fibers . . . . . . . . . . . .

219

5

A Brief History of Fiber-Optic Soliton Transmission . . . . . . . . . . . . . Fedor Mitschke

221

6

Perturbations of Solitons in Optical Fibers . . . . . . . . . . . . . . . . . . . . . Theodoros P. Horikis and Dimitrios J. Frantzeskakis

269

7

Emission of Dispersive Waves from Solitons in Axially Varying Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Kudlinski, A. Mussot, Matteo Conforti, and D. V. Skryabin

8

Nonlinear Waves in Multimode Fibers . . . . . . . . . . . . . . . . . . . . . . . . . I. S. Chekhovskoy, O. S. Sidelnikov, A. A. Reduyk, A. M. Rubenchik, O. V. Shtyrina, M. P. Fedoruk, S. K. Turitsyn, E. A. Zlobina, S. I. Kablukov, S. A. Babin, K. Krupa, V. Couderc, A. Tonello, A. Barthélémy, G. Millot, and S. Wabnitz

301 317

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Contents

9

Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stefano Trillo and Matteo Conforti

10

A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boris A. Malomed

Part III 11

Optical Fiber Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Advanced Nano-engineered Glass-Based Optical Fibers for Photonics Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. C. Paul, S. Das, A. Dhar, D. Dutta, P. H. Reddy, M. Pal, and A. V. Kir’yanov

12

Fabrication of Negative Curvature Hollow Core Fiber . . . . . . . . . . . Muhammad Rosdi Abu Hassan

13

Optimized Fabrication of Thulium Doped Silica Optical Fiber Using MCVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Z. Muhamad Yassin, Nasr Y. M. Omar, and Hairul Azhar Bin Abdul Rashid

373

421

475

477

529

551

14

Microfiber: Physics and Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . Horng Sheng Lin and Zulfadzli Yusoff

587

15

Flat Fibers: Fabrication and Modal Characterization . . . . . . . . . . . Ghafour Amouzad Mahdiraji, Katrina D. Dambul, Soo Yong Poh, and Faisal Rafiq Mahamd Adikan

623

16

3D Silica Lithography for Future Optical Fiber Fabrication . . . . . . Gang-Ding Peng, Yanhua Luo, Jianzhong Zhang, Jianxiang Wen, Yushi Chu, Kevin Cook, and John Canning

637

Part IV 17

18

Active Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

655

Rare-Earth-Doped Laser Fiber Fabrication Using Vapor Deposition Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sonja Unger, Florian Lindner, Claudia Aichele, and Kay Schuster

657

Powder Process for Fabrication of Rare Earth-Doped Fibers for Lasers and Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valerio Romano, Sönke Pilz, and Hossein Najafi

679

19

Progress in Mid-infrared Fiber Source Development . . . . . . . . . . . . Darren D. Hudson, Alexander Fuerbach, and Stuart D. Jackson

723

20

Crystalline Fibers for Fiber Lasers and Amplifiers . . . . . . . . . . . . . . Sheng-Lung Huang

757

Contents

21

22

Cladding-Pumped Multicore Fiber Amplifier for Space Division Multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kazi S. Abedin Optical Amplifiers for Mode Division Multiplexing . . . . . . . . . . . . . . Yongmin Jung, Shaif-Ul Alam, and David J. Richardson

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821 849

Volume 2 Part V Special Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

875

23

Optical Fibers for High-Power Lasers . . . . . . . . . . . . . . . . . . . . . . . . . Xia Yu, Biao Sun, Jiaqi Luo, and Elizabeth Lee

877

24

Multicore Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ming Tang

895

25

Polymer Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kishore Bhowmik and Gang-Ding Peng

967

26

Optical Fibers in Terahertz Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 1019 Georges Humbert

27

Optical Fibers for Biomedical Applications . . . . . . . . . . . . . . . . . . . . . 1069 Gerd Keiser

Part VI Optical Fiber Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097 28

Basics of Optical Fiber Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 1099 Mingjie Ding, Desheng Fan, Wenyu Wang, Yanhua Luo, and Gang-Ding Peng

29

Measurement of Active Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . . . 1139 Gui Xiao, Ghazal Fallah Tafti, Amirhassan Zareanborji, Anahita Ghaznavi, and Qiancheng Zhao

30

Characterization of Specialty Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . 1177 Quan Chai, Yushi Chu, and Jianzhong Zhang

31

Characterization of Distributed Birefringence in Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1227 Yongkang Dong, Lei Teng, Hongying Zhang, Taofei Jiang, and Dengwang Zhou

32

Characterization of Distributed Polarization-Mode Coupling for Fiber Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1259 Jun Yang, Zhangjun Yu, and Libo Yuan

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Contents

Part VII

Optical Fiber Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1299

33

Materials Development for Advanced Optical Fiber Sensors and Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1301 Peter Dragic and John Ballato

34

Optoelectronic Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335 Lei Wei

35

Fiber Grating Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1351 Christophe Caucheteur and Tuan Guo

36

CO2 -Laser-Inscribed Long Period Fiber Gratings: From Fabrication to Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1379 Yiping Wang and Jun He

37

Micro-/Nano-optical Fiber Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425 Fei Xu

Part VIII

Optical Fiber Device Measurement . . . . . . . . . . . . . . . . . . . . . 1465

38

Measurement of Optical Fiber Grating . . . . . . . . . . . . . . . . . . . . . . . . 1467 Zhiqiang Song, Jian Guo, Haifeng Qi, and Weitao Wang

39

Measurement of Optical Fiber Amplifier . . . . . . . . . . . . . . . . . . . . . . . 1485 Yanhua Luo, Binbin Yan, Jianxiang Wen, Jianzhong Zhang, and Gang-Ding Peng

40

Measurement of Optical Fiber Laser . . . . . . . . . . . . . . . . . . . . . . . . . . 1529 Haifeng Qi, Weitao Wang, Jian Guo, and Zhiqiang Song

Volume 3 Part IX

Distributed Optical Fiber Sensing . . . . . . . . . . . . . . . . . . . . . . . . 1557

41

Distributed Rayleigh Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1559 Xinyu Fan

42

Distributed Raman Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1609 Marcelo A. Soto and Fabrizio Di Pasquale

43

Distributed Brillouin Sensing: Time-Domain Techniques . . . . . . . . . 1663 Marcelo A. Soto

44

Distributed Brillouin Sensing: Frequency-Domain Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1755 Aleksander Wosniok

Contents

45

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Distributed Brillouin Sensing: Correlation-Domain Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1781 Weiwen Zou, Xin Long, and Jianping Chen

Part X Optical Fiber Sensors for Industrial Applications . . . . . . . . . . . 1813 46

Optical Fiber Sensors for Remote Condition Monitoring of Industrial Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1815 Tong Sun OBE, M. Fabian, Y. Chen, M. Vidakovic, S. Javdani, K. T. V. Grattan, J. Carlton, C. Gerada, and L. Brun

47

Optical Fiber Sensor Network and Industrial Applications . . . . . . . 1839 Qizhen Sun, Zhijun Yan, Deming Liu, and Lin Zhang

48

Fiber Optic Sensors for Coal Mine Hazard Detection . . . . . . . . . . . . 1885 Tongyu Liu

49

Optical Fiber Sensors in Ionizing Radiation Environments . . . . . . . 1913 Dan Sporea

Part XI Polymer Optical Fiber Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . 1955 50

Polymer Optical Fiber Sensors and Devices . . . . . . . . . . . . . . . . . . . . 1957 Ricardo Oliveira, Filipa Sequeira, Lúcia Bilro, and Rogério Nogueira

51

Solid Core Single-Mode Polymer Fiber Gratings and Sensors . . . . . 1997 Kishore Bhowmik, Gang Ding Peng, Eliathamby Ambikairajah, and Ginu Rajan

52

Microstructured Polymer Optical Fiber Gratings and Sensors . . . . 2037 Getinet Woyessa, Andrea Fasano, and Christos Markos

53

Polymer Fiber Sensors for Structural and Civil Engineering Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2079 Sascha Liehr

Part XII Photonic Crystal Fiber Sensing . . . . . . . . . . . . . . . . . . . . . . . . . 2115 54

Photonic Microcells for Sensing Applications . . . . . . . . . . . . . . . . . . . 2117 Chao Wang, Wei Jin, Hoi Lut Ho, and Fan Yang

55

Filling Technologies of Photonic Crystal Fibers and Their Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2139 Chun-Liu Zhao, D. N. Wang, and Limin Xiao

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Contents

56

Photonic Crystal Fiber-Based Grating Sensors . . . . . . . . . . . . . . . . . 2201 Changrui Liao, Feng Zhu, and Chupao Lin

57

Photonic Crystal Fiber-Based Interferometer Sensors . . . . . . . . . . . 2231 Min Wang, Jiankun Peng, Weijia Wang, and Minghong Yang

Part XIII

Optical Fiber Microfluidic Sensors . . . . . . . . . . . . . . . . . . . . . . 2281

58

Optical Fiber Microfluidic Sensors Based on Opto-physical Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2283 Chen-Lin Zhang, Chao-Yang Gong, Yuan Gong, Yun-Jiang Rao, and Gang-Ding Peng

59

Micro-/Nano-optical Fiber Microfluidic Sensors . . . . . . . . . . . . . . . . 2319 Lei Zhang

60

All Optical Fiber Optofluidic or Ferrofluidic Microsensors Fabricated by Femtosecond Laser Micromachining . . . . . . . . . . . . . 2351 Hai Xiao, Lei Yuan, Baokai Cheng, and Yang Song

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2389

About the Editor

Gang-Ding Peng Photonics and Optical Communications School of Electrical Engineering and Telecommunications University of New South Wales Sydney, NSW, Australia Gang-Ding Peng received his B.Sc. degree in physics from Fudan University, Shanghai, China, in 1982, and the M.Sc. degree in applied physics and Ph.D. in electronic engineering from Shanghai Jiao Tong University, Shanghai, China, in 1984 and 1987, respectively. From 1987 through 1988 he was a lecturer at Jiao Tong University. He was a postdoctoral research fellow in the Optical Sciences Centre of the Australian National University, Canberra, from 1988 to 1991. He has been working at the University of NSW in Sydney, Australia, since 1991; was a Queen Elizabeth II Fellow from 1992 to 1996; and is currently a professor in the same university. He is a fellow and life member of both Optical Society of America (OSA) and The International Society for Optics and Photonics (SPIE). His research interests include silica and polymer optical fibers, optical fiber and waveguide devices, optical fiber sensors, and nonlinear optics. He has worked in research and teaching in photonics and fiber optics for more than 30 years and maintained a high research profile internationally.

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Section Editors

Part I: Optical Fiber Communication

Ming-Jun Li

Corning Incorporated, Corning, NY, USA

Chao LU Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hong Kong SAR, China xv

xvi

Section Editors

Part II: Solitons and Nonlinear Waves in Optical Fibers

Boris A. Malomed Faculty of Engineering, Department of Physical Electronics, School of Electrical Engineering, Tel Aviv University, Tel Aviv, Israel ITMO University, St. Petersburg, Russia

Part III: Optical Fiber Fabrication

Hairul Azhar Bin Abdul Rashid Faculty of Engineering, Multimedia University, Cyberjaya, Malaysia

Section Editors

xvii

Part IV: Active Optical Fibers

Kyunghwan Oh Department of Physics, Institute of Physics and Applied Physics, Yonsei University, Seoul, Republic of Korea

Part V: Special Optical Fibers

Perry Shum

Nanyang Technological University, Singapore, Singapore

xviii

Section Editors

Zhilin Xu Center for Gravitational Experiments, School of Physics, Huazhong University of Science and Technology, Wuhan, China

Part VI: Optical Fiber Measurement

Jianzhong Zhang Key Lab of In-fiber Integrated Optics, Ministry of Education, Harbin Engineering University, Harbin, China

Section Editors

xix

Part VII: Optical Fiber Devices

John Canning i nterdisciplinary Photonics Laboratories (i PL), Global Big Data Technologies Centre (GBDTC), Tech Lab, School of Electrical and Data Engineering, University of Technology Sydney, Sydney, NSW, Australia

Tuan Guo

Institute of Photonics Technology, Jinan University, Guangzhou, China

xx

Section Editors

Part VIII: Optical Fiber Device Measurement

Yanhua Luo Photonics and Optical Communications, School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW, Australia Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province, Shenzhen University, Shenzhen, China

Part IX: Distributed Optical Fiber Sensing

Yosuke Mizuno Institute of Innovative Research, Tokyo Institute of Technology, Yokohama, Japan

Section Editors

xxi

Part X: Optical Fiber Sensors for Industrial Applications

Tong Sun OBE School of Mathematics, Computer Science and Engineering, City, University of London, London, UK

Part XI: Polymer Optical Fiber Sensing

Ginu Rajan School of Electrical, Computer and Telecommunications Engineering, University of Wollongong, Wollongong, Australia School of Electrical Engineering and Telecommunications, UNSW, Sydney, Australia

xxii

Section Editors

Part XII: Photonic Crystal Fiber Sensing

D. N. Wang College of Optical and Electrical Technology, China Jiliang University, Hangzhou, China

Part XIII: Optical Fiber Microfluidic Sensors

Yuan Gong Key Laboratory of Optical Fiber Sensing and Communications (Ministry of Education of China), University of Electronic Science and Technology of China, Chengdu, Sichuan, China

Contributors

John S. Abbott Corning Incorporated, Corning, NY, USA Hairul Azhar Bin Abdul Rashid Faculty of Engineering, Multimedia University, Cyberjaya, Malaysia Kazi S. Abedin OFS Laboratories, Somerset, NJ, USA Muhammad Rosdi Abu Hassan Centre for Optical Fibre Technology (COFT), School of Electrical, Electronic Engineering, Nanyang Technological University, Singapore, Singapore, Singapore Claudia Aichele Department of Fiber Optics, Leibniz Institute of Photonic Technology (Leibniz IPHT), Jena, Germany Shaif-Ul Alam Optoelectronics Research Centre (ORC), University of Southampton, Southampton, UK Eliathamby Ambikairajah School of Electrical Engineering and Telecommunications, UNSW, Sydney, Australia Ghafour Amouzad Mahdiraji School of Engineering, Taylor’s University, Subang Jaya, Selangor, Malaysia Flexilicate Sdn. Bhd., University of Malaya, Kuala Lumpur, Malaysia S. A. Babin Institute of Automation and Electrometry SB RAS, Novosibirsk, Russia Novosibirsk State University, Novosibirsk, Russia John Ballato Center for Optical Materials Science and Engineering Technologies (COMSET) and the Department of Materials Science and Engineering, Clemson University, Clemson, SC, USA A. Barthélémy XLIM, UMR CNRS 7252, Université de Limoges, Limoges, France Kishore Bhowmik HFC Assurance, Operate and Maintain Network, NBN, Melbourne, VIC, Australia Scott R. Bickham Corning Incorporated, Corning, NY, USA xxiii

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Contributors

Lúcia Bilro Instituto de Telecomunicações, Campus Universitário de Santiago, Aveiro, Portugal L. Brun Faiveley Brecknell Willis, Somerset, UK John Canning i nterdisciplinary Photonics Laboratories (i PL), Global Big Data Technologies Centre (GBDTC), Tech Lab, School of Electrical and Data Engineering, University of Technology Sydney, Sydney, NSW, Australia J. Carlton City, University of London, London, UK Christophe Caucheteur Electromagnetism and Telecommunication Department, University of Mons, Mons, Belgium Quan Chai Key Laboratory of In-Fiber Integrated Optics, Ministry Education of China, Harbin Engineering University, Harbin, China I. S. Chekhovskoy Novosibirsk State University, Novosibirsk, Russia Institute of Computational Technologies SB RAS, Novosibirsk, Russia Jianping Chen State Key Laboratory of Advanced Optical Communication Systems and Networks, Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China Xin Chen Corning Incorporated, Corning, NY, USA Y. Chen City, University of London, London, UK Baokai Cheng Department of Electrical and Computer Engineering, Center for Optical Materials Science and Engineering Technologies (COMSET), Clemson University, Clemson, SC, USA Yushi Chu Key Laboratory of In-Fiber Integrated Optics, Ministry Education of China, Harbin Engineering University, Harbin, China Photonics and Optical Communications, School of Electrical Engineering and Telecommunications, UNSW, Sydney, NSW, Australia i nterdisciplinary Photonics Laboratories (i PL), Global Big Data Technologies Centre (GBDTC), Tech Lab, School of Electrical and Data Engineering, University of Technology Sydney, Sydney, NSW, Australia J. Doug Coleman Corning Incorporated, Corning, NY, USA Matteo Conforti CNRS, UMR 8523 – PhLAM – Physique des Lasers Atomes et Molécules, University of Lille, Lille, France Kevin Cook i nterdisciplinary Photonics Laboratories (i PL), Global Big Data Technologies Centre (GBDTC), Tech Lab, School of Electrical and Data Engineering, University of Technology Sydney, Sydney, NSW, Australia V. Couderc XLIM, UMR CNRS 7252, Université de Limoges, Limoges, France

Contributors

xxv

Katrina D. Dambul Faculty of Engineering, Multimedia University, Cyberjaya, Selangor, Malaysia S. Das Fiber Optics and Photonics Division, CSIR-Central Glass and Ceramic Research Institute, Kolkata, India A. Dhar Fiber Optics and Photonics Division, CSIR-Central Glass and Ceramic Research Institute, Kolkata, India Mingjie Ding Photonics and Optical Communications, School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW, Australia Fabrizio Di Pasquale Institute of Communication, Information and Perception Technologies (TECIP), Scuola Superiore Sant’Anna, Pisa, Italy Yongkang Dong National Key Laboratory of Science and Technology on Tunable Laser, Harbin Institute of Technology, Harbin, China John D. Downie Corning Incorporated, Corning, NY, USA Peter Dragic Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, USA D. Dutta Fiber Optics and Photonics Division, CSIR-Central Glass and Ceramic Research Institute, Kolkata, India M. Fabian City, University of London, London, UK Ghazal Fallah Tafti Photonics and Optical Communications, School of Electrical Engineering and Telecommunications, UNSW, Sydney, NSW, Australia Desheng Fan Photonics and Optical Communications, School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW, Australia Xinyu Fan State Key Laboratory of Advanced Optical Communication Systems and Networks, Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China Andrea Fasano DTU Mekanik, Department of Mechanical Engineering, Technical University of Denmark, Lyngby, Denmark M. P. Fedoruk Novosibirsk State University, Novosibirsk, Russia Institute of Computational Technologies SB RAS, Novosibirsk, Russia Dimitrios J. Frantzeskakis Department of Physics, National and Kapodistrian University of Athens, Athens, Greece Alexander Fuerbach MQ Photonics Research Centre, Department of Physics and Astronomy, Macquarie University, North Ryde, NSW, Australia C. Gerada The University of Nottingham, Nottingham, UK

xxvi

Contributors

Anahita Ghaznavi Photonics and Optical Communications, School of Electrical Engineering and Telecommunications, UNSW, Sydney, NSW, Australia Chao-Yang Gong Key Laboratory of Optical Fiber Sensing and Communications (Ministry of Education of China), University of Electronic Science and Technology of China, Chengdu, Sichuan, China Yuan Gong Key Laboratory of Optical Fiber Sensing and Communications (Ministry of Education of China), University of Electronic Science and Technology of China, Chengdu, Sichuan, China K. T. V. Grattan City, University of London, London, UK Jian Guo Shandong Key Laboratory of Optical Fiber Sensing Technologies, Qilu Industry University (Laser Institute of Shandong Academy of Sciences), Jinan, China Tuan Guo Institute of Photonics Technology, Jinan University, Guangzhou, China Tetsuya Hayashi Optical Communications Laboratory, Sumitomo Electric Industries, Ltd., Yokohama, Kanagawa, Japan Jun He Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province, College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen, China Guangdong and Hong Kong Joint Research Centre for Optical Fibre Sensors, Shenzhen University, Shenzhen, China Hoi Lut Ho Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong, China Theodoros P. Horikis Department of Mathematics, University of Ioannina, Ioannina, Greece Sheng-Lung Huang Graduate Institute of Photonics and Optoelectronics, and Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan Darren D. Hudson MQ Photonics Research Centre, Department of Physics and Astronomy, Macquarie University, North Ryde, NSW, Australia Georges Humbert XLIM Research Institute, UMR 7252 CNRS, University of Limoges, Limoges, France Ezra Ip NEC Laboratories America, Princeton, NJ, USA Stuart D. Jackson Department of Engineering, MQ Photonics Research Centre, School of Engineering, Macquarie University, North Ryde, NSW, Australia S. Javdani City, University of London, London, UK Taofei Jiang National Key Laboratory of Science and Technology on Tunable Laser, Harbin Institute of Technology, Harbin, China

Contributors

xxvii

Wei Jin Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong, China Yongmin Jung Optoelectronics Research Centre (ORC), University of Southampton, Southampton, UK S. I. Kablukov Institute of Automation and Electrometry SB RAS, Novosibirsk, Russia Gerd Keiser Boston University, Boston, MA, USA A. V. Kir’yanov Centro de Investigaciones en Optica, Guanajuato, Mexico K. Krupa Department of Information Engineering, University of Brescia, Brescia, Italy A. Kudlinski CNRS, UMR 8523 – PhLAM – Physique des Lasers Atomes et Molécules, University of Lille, Lille, France Elizabeth Lee Precision Measurements Group, Singapore Institute of Manufacturing Technology, Singapore, Singapore Ming-Jun Li Corning Incorporated, Corning, NY, USA Changrui Liao College of Optoelectronic Engineering, Shenzhen University, Shenzhen, China Sascha Liehr Division 8.6 “Fibre Optic Sensors”, Bundesanstalt für Materialforschung und –prüfung (BAM), Berlin, Germany Chupao Lin College of Optoelectronic Engineering, Shenzhen University, Shenzhen, China Horng Sheng Lin Universiti Tunku Abdul Rahman, Sungai Long Campus, Kajang, Malaysia Florian Lindner Department of Fiber Optics, Leibniz Institute of Photonic Technology (Leibniz IPHT), Jena, Germany Deming Liu School of Optical and Electronic Information, Next Generation Internet Access National Engineering Laboratory (NGIAS), Huazhong University of Science and Technology, Wuhan, Hubei, P. R. China Tongyu Liu Laser Institute, Qilu University of Technology-Shandong Academy of Science, Jinan, Shandong, China Xin Long State Key Laboratory of Advanced Optical Communication Systems and Networks, Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China Jiaqi Luo Precision Measurements Group, Singapore Institute of Manufacturing Technology, Singapore, Singapore

xxviii

Contributors

Yanhua Luo Photonics and Optical Communications, School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW, Australia Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province, Shenzhen University, Shenzhen, China Faisal Rafiq Mahamd Adikan Flexilicate Sdn. Bhd., University of Malaya, Kuala Lumpur, Malaysia Integrated Lightwave Research Group, Department of Electrical Engineering, Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia Sergejs Makovejs Corning Incorporated, Ewloe, UK Boris A. Malomed Faculty of Engineering, Department of Physical Electronics, School of Electrical Engineering, Tel Aviv University, Tel Aviv, Israel ITMO University, St. Petersburg, Russia Christos Markos DTU Fotonik, Department of Photonics Engineering, Technical University of Denmark, Lyngby, Denmark G. Millot ICB, UMR CNRS 6303, Université de Bourgogne, Dijon, France Fedor Mitschke Institut für Physik, Universität Rostock, Rostock, Germany S. Z. Muhamad Yassin Photonics Laboratory, Telekom Research and Development, Cyberjaya, Malaysia A. Mussot CNRS, UMR 8523 – PhLAM – Physique des Lasers Atomes et Molécules, University of Lille, Lille, France Hossein Najafi Institute for Applied Laser, Photonics and Surface Technologies (ALPS), Bern University of Applied Sciences, Burgdorf, Switzerland Rogério Nogueira Instituto de Telecomunicações, Campus Universitário de Santiago, Aveiro, Portugal Ricardo Oliveira Instituto de Telecomunicações, Campus Universitário de Santiago, Aveiro, Portugal Nasr Y. M. Omar Faculty of Engineering, Multimedia University, Cyberjaya, Malaysia M. Pal Fiber Optics and Photonics Division, CSIR-Central Glass and Ceramic Research Institute, Kolkata, India M. C. Paul Fiber Optics and Photonics Division, CSIR-Central Glass and Ceramic Research Institute, Kolkata, India Gang-Ding Peng Photonics and Optical Communications, School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW, Australia

Contributors

xxix

Jiankun Peng National Engineering Laboratory for Fiber Optic Sensing Technology (NEL-FOST), Wuhan University of Technology, Wuhan, China Sönke Pilz Institute for Applied Laser, Photonics and Surface Technologies (ALPS), Bern University of Applied Sciences, Burgdorf, Switzerland Soo Yong Poh Integrated Lightwave Research Group, Department of Electrical Engineering, Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia Haifeng Qi Shandong Key Laboratory of Optical Fiber Sensing Technologies, Qilu Industry University (Laser Institute of Shandong Academy of Sciences), Jinan, China Ginu Rajan School of Electrical, Computer and Telecommunications Engineering, University of Wollongong, Wollongong, Australia School of Electrical Engineering and Telecommunications, UNSW, Sydney, Australia Yun-Jiang Rao Key Laboratory of Optical Fiber Sensing and Communications (Ministry of Education of China), University of Electronic Science and Technology of China, Chengdu, Sichuan, China P. H. Reddy Academy of Scientific and Innovative Research (AcSIR), IR-CGCRI Campus, Kolkata, India A. A. Reduyk Novosibirsk State University, Novosibirsk, Russia David J. Richardson Optoelectronics Research Centre (ORC), University of Southampton, Southampton, UK Valerio Romano Institute for Applied Laser, Photonics and Surface Technologies (ALPS), Bern University of Applied Sciences, Burgdorf, Switzerland Institute of Applied Physics (IAP), University of Bern, Bern, Switzerland A. M. Rubenchik Lawrence Livermore National Laboratory, Livermore, CA, USA Kay Schuster Department of Fiber Optics, Leibniz Institute of Photonic Technology (Leibniz IPHT), Jena, Germany Filipa Sequeira Instituto de Telecomunicações, Campus Universitário de Santiago, Aveiro, Portugal O. V. Shtyrina Novosibirsk State University, Novosibirsk, Russia Institute of Computational Technologies SB RAS, Novosibirsk, Russia O. S. Sidelnikov Novosibirsk State University, Novosibirsk, Russia D. V. Skryabin Department of Nanophotonics and Metamaterials, ITMO University, St Petersburg, Russia Department of Physics, University of Bath, Bath, UK

xxx

Contributors

Yang Song Department of Electrical and Computer Engineering, Center for Optical Materials Science and Engineering Technologies (COMSET), Clemson University, Clemson, SC, USA Zhiqiang Song Shandong Key Laboratory of Optical Fiber Sensing Technologies, Qilu Industry University (Laser Institute of Shandong Academy of Sciences), Jinan, China Marcelo A. Soto Institute of Electrical Engineering, EPFL Swiss Federal Institute of Technology, Lausanne, Switzerland Dan Sporea National Institute for Laser, Plasma and Radiation Physics, Center for Advanced Laser Technologies, M˘agurele, Romania Biao Sun Precision Measurements Group, Singapore Institute of Manufacturing Technology, Singapore, Singapore Qizhen Sun School of Optical and Electronic Information, Next Generation Internet Access National Engineering Laboratory (NGIAS), Huazhong University of Science and Technology, Wuhan, Hubei, P. R. China Tong Sun OBE School of Mathematics, Computer Science and Engineering, City, University of London, London, UK Ming Tang Wuhan National Lab for Optoelectronics (WNLO) and National Engineering Laboratory for Next Generation Internet Access System (NGIA), School of Optical and Electronic Information, Huazhong University of Science and Technology (HUST), Wuhan, China Lei Teng National Key Laboratory of Science and Technology on Tunable Laser, Harbin Institute of Technology, Harbin, China A. Tonello XLIM, UMR CNRS 7252, Université de Limoges, Limoges, France Stefano Trillo Department of Engineering, University of Ferrara, Ferrara, Italy S. K. Turitsyn Novosibirsk State University, Novosibirsk, Russia Aston Institute of Photonic Technologies, Aston University, Birmingham, UK Sonja Unger Department of Fiber Optics, Leibniz Institute of Photonic Technology (Leibniz IPHT), Jena, Germany M. Vidakovic City, University of London, London, UK S. Wabnitz Novosibirsk State University, Novosibirsk, Russia Department of Information Engineering, University of Brescia, Brescia, Italy National Institute of Optics INO-CNR, Brescia, Italy Chao Wang School of Electrical Engineering, Wuhan University, Wuhan, Hubei, China D. N. Wang College of Optical and Electrical Technology, China Jiliang University, Hangzhou, China

Contributors

xxxi

Min Wang National Engineering Laboratory for Fiber Optic Sensing Technology (NEL-FOST), Wuhan University of Technology, Wuhan, China School of Electronic and Electrical Engineering, Wuhan Textile University, Wuhan, China Weijia Wang National Engineering Laboratory for Fiber Optic Sensing Technology (NEL-FOST), Wuhan University of Technology, Wuhan, China Weitao Wang Shandong Key Laboratory of Optical Fiber Sensing Technologies, Qilu Industry University (Laser Institute of Shandong Academy of Sciences), Jinan, China Wenyu Wang Photonics and Optical Communications, School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW, Australia Yiping Wang Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province, College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen, China Guangdong and Hong Kong Joint Research Centre for Optical Fibre Sensors, Shenzhen University, Shenzhen, China Lei Wei School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, Singapore Jianxiang Wen Key Laboratory of Specialty Fiber Optics and Optical Access Networks, Shanghai University, Shanghai, China Aleksander Wosniok 8.6 Fibre Optic Sensors, Federal Institute for Materials Research and Testing (BAM), Berlin, Germany Getinet Woyessa DTU Fotonik, Department of Photonics Engineering, Technical University of Denmark, Lyngby, Denmark Gui Xiao Photonics and Optical Communications, School of Electrical Engineering and Telecommunications, UNSW, Sydney, NSW, Australia Hai Xiao Department of Electrical and Computer Engineering, Center for Optical Materials Science and Engineering Technologies (COMSET), Clemson University, Clemson, SC, USA Limin Xiao Advanced Fiber Devices and Systems Group, Key Laboratory of Micro and Nano Photonic Structures (MoE), Department of Optical Science and Engineering Fudan University, Shanghai, China Key Laboratory for Information Science of Electromagnetic Waves (MoE), Fudan University, Shanghai, China Shanghai Engineering Research Center of Ultra-Precision Optical Manufacturing, Fudan University, Shanghai, China

xxxii

Contributors

Fei Xu National Laboratory of Solid State Microstructures and College of Engineering and Applied Sciences, Nanjing University, Nanjing, Jinagsu, P. R. China Binbin Yan State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing, China Zhijun Yan School of Optical and Electronic Information, Next Generation Internet Access National Engineering Laboratory (NGIAS), Huazhong University of Science and Technology, Wuhan, Hubei, P. R. China Fan Yang Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong, China Jun Yang Key Lab of In-Fiber Integrated Optics, Ministry Education of China, Harbin Engineering University, Harbin, China College of Science, Harbin Engineering University, Harbin, China Minghong Yang National Engineering Laboratory for Fiber Optic Sensing Technology (NEL-FOST), Wuhan University of Technology, Wuhan, China Xia Yu Precision Measurements Group, Singapore Institute of Manufacturing Technology, Singapore, Singapore Zhangjun Yu Key Lab of In-Fiber Integrated Optics, Ministry Education of China, Harbin Engineering University, Harbin, China College of Science, Harbin Engineering University, Harbin, China Lei Yuan Department of Electrical and Computer Engineering, Center for Optical Materials Science and Engineering Technologies (COMSET), Clemson University, Clemson, SC, USA Libo Yuan Key Lab of In-Fiber Integrated Optics, Ministry Education of China, Harbin Engineering University, Harbin, China College of Science, Harbin Engineering University, Harbin, China Zulfadzli Yusoff Multimedia University, Persiaran Multimedia, Cyberjaya, Malaysia Amirhassan Zareanborji Photonics and Optical Communications, School of Electrical Engineering and Telecommunications, UNSW, Sydney, NSW, Australia Chen-Lin Zhang Key Laboratory of Optical Fiber Sensing and Communications (Ministry of Education of China), University of Electronic Science and Technology of China, Chengdu, Sichuan, China Hongying Zhang Institute of Photonics and Optical Fiber Technology, Harbin University of Science and Technology, Harbin, China Jianzhong Zhang Key Lab of In-fiber Integrated Optics, Ministry of Education, Harbin Engineering University, Harbin, China

Contributors

xxxiii

Lei Zhang College of Optical Science and Engineering, Zhejiang University, Hangzhou, China Lin Zhang Aston Institute of Photonic Technologies, Aston University, Birmingham, UK Chun-Liu Zhao College of Optical and Electrical Technology, China Jiliang University, Hangzhou, China Qiancheng Zhao Photonics and Optical Communications, School of Electrical Engineering and Telecommunications, UNSW, Sydney, NSW, Australia Dengwang Zhou National Key Laboratory of Science and Technology on Tunable Laser, Harbin Institute of Technology, Harbin, China Feng Zhu College of Optoelectronic Engineering, Shenzhen University, Shenzhen, China E. A. Zlobina Institute of Automation and Electrometry SB RAS, Novosibirsk, Russia Weiwen Zou State Key Laboratory of Advanced Optical Communication Systems and Networks, Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China

Part I Optical Fiber for Communication

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Single-Mode Fibers for High Speed and Long-Haul Transmission John D. Downie, Ming-Jun Li, and Sergejs Makovejs

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background and History of Optical Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . History of Fiber Evolution (1966–1987) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . History of Fiber Evolution (1987–2007) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . History of Fiber Evolution (2007 Onwards) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optical Fiber Designs for Long-Haul Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantification of System Level Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Long-Haul and Ultra-Long-Haul Transmission Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . Raman Gain Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unrepeatered Span Transmission Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transmission System Modeling and Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Factors and Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Splice Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Practical Benefits of Ultra-Low Attenuation and Large Effective Area Fibers . . . . . . . . . . . Potential Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 5 5 6 8 11 16 17 20 23 27 30 30 31 35 36 38

Abstract

The design and manufacture of optical fibers have evolved over time as optical system technologies and data rates have changed. Fiber characteristics and parameters that were important for previous system generations may be different

J. D. Downie () · M.-J. Li Corning Incorporated, Corning, NY, USA e-mail: [email protected]; [email protected] S. Makovejs Corning Incorporated, Ewloe, UK e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 G.-D. Peng (ed.), Handbook of Optical Fibers, https://doi.org/10.1007/978-981-10-7087-7_65

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now in the era of coherent transmission systems with data rates of 100 Gb/s and higher for long-haul (LH) and ultra-long-haul (ULH) transmission links. New systems are designed with no in-line optical dispersion compensation so that all dispersion compensation is performed digitally in the transmitter and receiver. In this system approach, attenuation and nonlinear tolerance are the fiber characteristics that have the largest impact on overall system performance. In this chapter, we examine the history of single-mode fiber designs and quantify differences in performance of various fibers. This is done mainly in the context of conventional repeatered LH and ULH systems, with a brief consideration of the special case of long single-span unrepeatered systems. Practical aspects of different fibers are also considered, including bend performance and splice loss.

Introduction Fiber-optic communication systems are comprised of several separate, but interdependent, parts. The overall performance of a system depends on the quality and characteristics of each element. The transmitter and receiver equipment are active components of the system that determine the bit rate, modulation format, spectral efficiency, capacity, and other system aspects. A second important part of communication systems is optical amplification. The type of optical amplifiers, and the noise figure and spectral bandwidth of those amplifiers also impact the performance and total capacity that the system can carry. These devices are active parts of the system in the sense that they are electrically powered and may have active control of their operation and performance. A third major element of fiberoptic systems is the transmission medium, optical fiber, a purely passive part of the system. There are also other smaller passive components such as couplers, taps, optical multiplexers, and de-multiplexers, but these play a much smaller role in determining the performance, capacity, and reach of optical communication systems. In this chapter, we examine the properties of single-mode optical fibers that promote the best performance in modern coherent transmission systems. With respect to fiber, the highest or best system performance generally translates into longest reach before regeneration is required and largest total capacity that can be carried by a fiber over a given distance. We study the role of fiber characteristics here largely in this context of system performance, which can be evaluated in quantities such as Q-factor (inversely related to bit error rate or BER) and fiber figure of merit. We also touch on practical performance issues associated with optical fibers such as macro-bend loss, micro-bend loss, splice loss, and Raman gain. We will show that for today’s coherent systems, the fiber characteristics that have the greatest influence on overall system performance are attenuation and nonlinear tolerance, mainly governed by the fiber effective area. The rest of this chapter is organized in the following manner; in the first section, we first take a look at the evolution of optical fiber in a historical context and describe the four generations of optical transmission systems and the fibers used in

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them. In section “Optical Fiber Designs for Long-haul Transmission,” we discuss design aspects of optical fibers and the constraints and interdependencies of various fiber parameters such as dispersion, effective area, and cutoff wavelength. In the third section, we look at means to quantify system level performance with regard to fiber characteristics. This is done mainly with regard to long-haul systems with optical amplifiers at the end of each fiber span, but we briefly explore which fiber characteristics have most effect in unrepeatered span systems as well. Modeling and experimental data are shown, comparing the reach lengths of different singlemode optical fiber types under comparable system conditions and configurations. In the fourth section, splice loss considerations and issues are discussed, along with some other practical benefits that accrue from the use of high-performing fibers with low attenuation and large effective area. Finally, we mention one possible future direction of fibers that stretches the definition from single-mode to quasi-single-mode.

Background and History of Optical Fiber History of Fiber Evolution (1966–1987) The proposal of low loss silica-based optical fiber by Charles Kao in 1966 (Kao and Hockham 1966) marked the debut of optical communications. A major milestone in developing optical fibers was the demonstration of optical fiber with attenuation less than 20 dB/km in 1970 (Kapron et al. 1970), which first enabled the use of optical fibers for practical transmission applications. Since then, optical fiber, components, and transmission system technologies have advanced rapidly to increase the transmission capacity of fibers and cables during the past five decades. In fact, the transmission capacity of a single fiber has increased by a factor of approximately ten every 4 years. The long-haul optical fiber transmission system has gone through four generations. The first fiber transmission system generation utilized multimode optical fibers and light emitting diode (LED) sources operating in the 850 nm wavelength region (Sanferrare 1987). Multimode fibers have a large core size and high numerical aperture, which facilitated coupling of the light from the LED source into the fiber with low cost. However, the transmission rate and distance afforded by multimode fiber-based systems are limited by the fundamental bandwidth limitation of multimode fibers due to intermodal dispersion. The solution to eliminating intermodal dispersion is to use a single-mode optical fiber, i.e., a fiber that supports only one transverse mode. With the development of semiconductor lasers (Joyce et al. 1977) and the opening of long wavelength transmission windows (Miya et al. 1979) as well as advances in single-mode fiber coupling technology (Tynes and Derosier 1977) in the later 1970s, single-mode fiber transmission systems became possible. The second generation of optical fiber communication systems employed standard single-mode fiber and single-mode lasers operating at 1310 nm. Standard single-mode fiber has lower attenuation than

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multimode fiber and exhibits nearly zero chromatic dispersion in the 1310 nm wavelength region, enabling longer transmission distance with higher data rates. Although the attenuation of a single-mode optical fiber is lowest in the 1550 nm wavelength window, the chromatic dispersion in this wavelength window is rather large (about C17 ps/km/nm) due to silica glass material dispersion. If uncompensated, dispersion of this order can be a limitation for high data rate systems. In order to overcome the dispersion limitation, dispersion-shifted optical fiber (DSF) was proposed (Cohen et al. 1979). In a dispersion shifted fiber, the material dispersion of silica glass is compensated by waveguide dispersion through refractive index profile designs, resulting in zero total dispersion at 1550 nm. This shift in fiber design allowed the use of conventional lasers exhibiting relatively large spectral width of several nm, which enabled the third generation optical fiber transmission system operating at 1550 nm. The dispersion shifted fiber was designed for single wavelength transmission at 1550 nm before multi-wavelength transmission systems were considered.

History of Fiber Evolution (1987–2007) In the late 1980s, the fourth generation high capacity optical fiber transmission system was driven by the development of the erbium doped fiber amplifier (EDFA) (Desurvire et al. 1987) and wavelength-division multiplexing (WDM) (Taga et al. 1988). WDM technology allows the simultaneous transmission of multiple wavelengths in a single fiber. For WDM transmission, dispersion shifted fibers proved unsuitable because the crosstalk between two neighboring channels (due to the nonlinear effect of four-wave mixing) is strongest when fiber dispersion is zero. The four-wave mixing effect can be suppressed effectively with a certain amount of chromatic dispersion. On the other hand, the dispersion should be small enough to minimize pulse broadening. Considering nonlinear effects, dispersion, and their interplay, a new type of fiber known as non-zero dispersion-shifted fiber (NZDSF) was proposed (Tkach et al. 1995). Non-zero dispersion-shifted fibers have typical dispersion values of 3–8 ps/nm/km at 1550 nm with effective areas of about 50 m2 . Because nonlinear effects and impairments are proportional to power density in the core (power divided by the effective area), fibers with larger effective areas are beneficial in terms of reducing the nonlinear effects. To increase the effective area, index profile designs with large effective area were proposed and NZDSF fiber with an effective area of about 72 m2 was developed for WDM transmission (Liu 1997). NZDSFs have since been widely deployed worldwide for high capacity WDM networks, in large part for systems with 10 Gb/s channel data rates. Wavelength division multiplexing technology added a new dimension to increase the transmission capacity of optical fiber and has been an ingrained feature of fiber communication systems since the 1990s. In conjunction with WDM development, channel rates have increased to meet rising traffic demands. This was largely made possible through bandwidth improvements of electrical and opto-electronic

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Fig. 1 Impact of noise on signal quality and an error in bit detection

devices such as modulators, photodiodes, and RF components. The channel rate of long-haul systems increased from 2.5 to 10 Gb/s using intensity modulation and direct detection. These systems were based on amplitude modulation, in which the information is encoded as either “1” or “0,” and the receiver uses a direct detection technique to determine whether “1” or “0” was transmitted. Under normal circumstances with high quality optical signals, the receiver can easily differentiate “1” from “0” on the other side of the transmission line. However, when the accumulated noise on a signal is high (e.g., very long transmission links with a series of optical amplifiers), a receiver is more likely to make a detection error. For example, an error can result from the receiver’s interpretation that the transmitted bit was “0,” when in reality the transmitted bit was “1” as illustrated in Fig. 1. A primary feature and advantage of systems based on direct detection is simplicity of implementation. However, the transmission bit-rate is not easily scaled. In such systems, an increase in bit rate is achieved through an increase in the time-division-multiplexed (TDM) rate and requires shorter transmission time slots and wider signal spectral widths. In the 1990s and 2000s, a significant amount of research was devoted to investigating the performance and practicality of >10 Gb/s transmission systems based on electrical and optical time division multiplexing (OTDM and ETDM) (Makovejs et al. 2008; Turkiewicz et al. 2005). Since an increase in the TDM rate also increases the spectral width, it does not necessarily enable a substantial increase in spectral efficiency and capacity per fiber. In addition, the move to 40 Gb/s intensity-modulated systems made evident a new range of significant intra-channel nonlinear effects that had to be managed. For several reasons, 10 Gb/s WDM systems were the last mass-adopted systems that employed direct detection; the success of 40 Gb/s direct-detection systems was limited and short-lived. Another notable feature of 2.5–10 Gb/s direct-detection long-haul systems is that they required in-line optical dispersion compensation modules in order to realize received signals with low accumulated chromatic dispersion. These compensation modules were periodically deployed at amplification sites in the mid-stage of 2-stage EDFAs. The compensation technology was largely based on either highly negative dispersion compensating fiber (DCF), or dispersion compensating gratings (DCGs). Both solutions introduced an additional loss and therefore an additional amplifier. Thus, the use of low-dispersion transmission fibers in terrestrial systems was beneficial in terms of reducing the total number of dispersion compensation modules, dual-stage amplifiers, and total link loss.

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Fig. 2 Dispersion compensation in previous-generation submarine systems. (a) Schematic diagram; (b) Dispersion map

While both terrestrial and submarine optical fiber transmission systems obey similar rules of physics, a different approach to dispersion compensation was pursued in direct detection submarine systems operating at bit rates such as 10 Gb/s. One of the most popular approaches relied on splicing (and subsequently re-coating) positive and negative dispersion fibers within the same span to achieve dispersion management. In the type of configuration shown in Fig. 2, hybrid spans comprised of both positive and negative dispersion fibers had a small net negative dispersion. After N hybrid fiber spans, the next span was made from only positive dispersion fiber to bring the total dispersion back to near zero. The larger section of N hybrid spans followed by one positive dispersion span was repeated M times to form the whole submarine cable.

History of Fiber Evolution (2007 Onwards) In the late 2000s, the first commercial 100 Gb/s systems with coherent detection were developed and became commercially available. The advent of coherent receiver technology revolutionized the way the industry approached the design of optical networks. Coherent detection allowed information to be encoded not only in amplitude, but also in phase and state of polarization, thereby increasing the number of bits per symbol period and the amount of information encoded per channel. For example, 100 Gb/s systems (total bit rate in the range of 112–128 Gb/s including forward error correction overhead) are now based on polarization-multiplexed quadrature phase shift keying (PM-QPSK). With TDM symbol rates of only 28–32 Gbaud, the use of QPSK modulation encodes 2 bits of information per symbol, and transmitting independent QPSK data on two orthogonal polarizations produces an overall bit rate multiplier of 4, resulting in bit rates of 112–128 Gb/s.

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Such spectrally efficient increases of the overall bit rate were made possible through integration of digital signal processing (DSP) functions in coherent receivers to create digital coherent receivers. Significant additional benefits of systems using digital coherent receivers include the ability to compensate for large amounts of chromatic dispersion and polarization mode dispersion in the receiver DSP, which has two important implications. First, the use of in-line dispersion compensation became superfluous and even undesirable from the standpoint of transmission performance. All-dispersive systems (sometimes called dispersion-uncompensated as compared to systems with periodic, in-line dispersion compensation) proved to be more resilient toward the impact of intra-channel and inter-channel nonlinear effects. In these systems, the accumulated chromatic dispersion is compensated digitally in the receiver DSP. Second, the ability to compensate for large amounts of polarization mode dispersion (PMD) in DSP (not previously possible with direct-detection systems) facilitates an upgrade in the TDM rate from 10 Gbaud to more than 25 Gbaud. In direct-detection systems with on-off keying modulation, upgrading to data rates higher than 10 Gb/s was largely limited by PMD (particularly pronounced for upgrades over legacy deployed optical fibers). Currently, 100 Gb/s coherent transmission systems now represent the workhorse of long-haul transmission systems deployed. As will be shown shortly, coherent technology also modified the requirements for optical fibers with a distinct shift toward lower loss and larger effective area. Another important fiber attribute that has gained attention in coherent systems is the nonlinear refractive index n2 , which, along with effective area, determines the n2 . For nonlinear tolerance of a fiber through the nonlinear coefficient  D 2  Aeff example, in traditional Germania-doped fibers the nonlinear index n2 is approximately 2.3  1020 m2 /W, although the exact value will be dependent on Germania concentration in the core. Large effective area fibers typically have slightly lower n2 due to lower Germania concentration in the core to achieve the required difference in refractive indices between the core and the cladding. The n2 can be reduced to approximately 2.1  1020 m2 /W (up to 10% reduction) by using silica-core fiber designs with Fluorine-doped cladding. In systems with coherent detection, chromatic dispersion (CD) also contributes to system performance. All other things being equal, non-zero dispersion shifted fibers (NZ-DSF) with CD equal to 4 ps/nm/km at 1550 nm (ITU-T G.655 2009) incur a transmission penalty compared to standard single-mode fibers with CD equal to 16 ps/nm/km at 1550 nm (ITU-T G.652 2016). This is explored in more detail later. The performance advantage of higher dispersion fibers is partly because they enable a more rapid spreading of the signal, causing faster reduction in signal peakto-mean ratio, and thereby improving tolerance toward nonlinear effects. The higher dispersion also promotes faster walkoff of channels with different wavelengths, decreasing interchannel nonlinear effects. A further increase in CD from 16 to 21 ps/nm/km (typical CD value for large effective area fibers) (ITU-T G.654 2016) will provide additional tolerance toward nonlinear effects, albeit the improvement will be less pronounced compared to an increase from 4 to 16 ps/nm/km.

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a

1.E-01 PM-16QAM PM-8QAM

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Fig. 3 (a) OSNR sensitivity of different modulation formats, (b) Normalized reach of each format

Overall, there seems to be a consensus within the industry that optical transport systems with coherent detection will dominate the landscape for the foreseeable future. The future bit-rates will continue to grow beyond 100 Gb/s through the use of even more spectrally efficient modulation (e.g., 8 or 16-state quadrature amplitude modulation) and probabilistic constellation shaping. As those spectrally efficient modulation formats have more stringent optical signal-to-noise ratio requirements, the importance of advanced fiber characteristics discussed previously will be even more pronounced for >100 Gb/s systems (particularly, for long-reach transmission). For example, consider Fig. 3a which shows the nominal OSNR sensitivity data (bit error rate BER vs. OSNR) for 32 Gbaud PM-QPSK, PM-8QAM, and PM-16QAM. The net data rate 150 Gb/s PM-8QAM signal requires about 4 dB higher OSNR than 100 Gb/s PM-QPSK, and 200 Gb/s PM-16QAM requires about 6.7 dB higher OSNR

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11

Fig. 4 Schematic illustration of the evolution of optical fiber transmission systems

than PM-QPSK. These higher OSNR requirements for higher spectral efficiency formats translate directly into shorter reach lengths for the same system. In Fig. 3b, the normalized reach lengths for the three formats are shown for a hypothetical system corresponding to the required OSNR values at a BER of about 1  103 . The reach lengths of PM-8QAM and PM-16QAM are reduced relative to PM-QPSK by about 60% and 80%, respectively. This reach reduction for higher spectral efficiency modulation formats illustrates the need for optical fiber improvements to help extend the possible reach of such formats to practical and cost-effective distances. The overall progression of technologies in terrestrial and submarine optical fiber transmission systems, along with key optical fiber characteristics, are summarized in Fig. 4.

Optical Fiber Designs for Long-Haul Transmission To design an optical fiber for long-haul transmission, there are several factors to consider such as attenuation, mode field diameter or effective area, cutoff wavelength, and chromatic dispersion. The total attenuation of an optical fiber is the sum of intrinsic and extrinsic attenuation. Figure 5 shows the major attenuation components of a generic silicabased optical fiber. Intrinsic attenuation is due to fundamental properties of glass materials used to construct the fiber core and cladding. Factors include Rayleigh scattering, and infrared and ultraviolet absorption tails in the transmission window. Extrinsic attenuation factors include absorption due to impurities such as transition metals and OH ions, small angle scattering (SAS) due to waveguide imperfections, and loss due to fiber bending effects. The most important intrinsic attenuation factor is Rayleigh scattering loss. Rayleigh scattering loss (˛ RS ) is the sum of scattering due to density fluctuations (˛  ) and scattering from concentration fluctuations (˛ c ). This is expressed as

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J. D. Downie et al.

Fig. 5 Major attenuation components for a silica-based optical fiber

˛RS D ˛ C ˛c ;

(1)

where ˛  is given by ˛ D

8 3 8 2 n p ˇ T kB T f ; 34

(2)

Tf is fictive temperature,  is incident wavelength, p is the photoelastic coefficient, n is refractive index, kB is the Boltzmann constant, and ˇ T is isothermal compressibility (Maksimov et al. 2011). Of these, the most important parameter is glass fictive temperature, the temperature where glass structure is the same as that of a supercooled liquid. The concentration fluctuation scattering loss is proportional to the gradient of dopant concentration C as  ˛c 

@n @C

2

˝ ˛ C 2 Tf

(3)

Because both the density and concentration fluctuation components of Rayleigh scattering are proportional to the fictive temperature, it is important to reduce Tf as much as possible to increase structural relaxation. To further reduce the concentration fluctuation, it is advantageous to reduce the dopant level in the core (Lines 1994; Kakiuchida et al. 2003).

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Fig. 6 Profile designs for large effective area fibers

The aforementioned impurities which contribute to extrinsic attenuation can be mostly addressed and eliminated in the chemical vapor deposition process utilized in fiber production. Scattering losses due to waveguide imperfections can be minimized by matching core and cladding glass viscosity and using advanced fiber-manufacturing technology. Extrinsic attenuation due to both macro- and micro-bending losses is primarily addressed in fiber design. Optical fiber properties such as effective area, cutoff wavelength, chromatic dispersion, and bending loss are governed by refractive index profile design. As mentioned earlier, fiber chromatic dispersion was a very important fiber parameter for direct detection systems due to transmission limiting pulse broadening. However, with new developments in coherent detection and digital signal processing technologies, transmission impairments from fiber dispersion can be compensated digitally. Low dispersion is no longer a factor that needs to be considered in fiber design. In fact, high dispersion is desirable because it can reduce nonlinear effects. Due to the reduced impact of dispersion, the most influential fiber parameters are effective area, cable cutoff wavelength, and attenuation due to bending losses. Figure 6 shows three general profile design types that can be used for fibers with large effective area and low attenuation. Figure 6a is a step index profile design. This simple design has two profile parameters: the relative refractive index change (core delta) and the core radius. To increase the effective area, the core radius can be increased but core delta must be reduced to keep the cable cutoff wavelength below the minimum wavelength of the application window, e.g., 1530 nm for 1550 nm window. Due to the cable cutoff wavelength specification limitations in international standards, the bend losses will tend to increase when the effective area gets larger. In general, there are system-based specifications for attenuation due to macrobending, e.g., less than 0.5 dB loss for 100 turns at a bend diameter of 60 mm. For a step index design with cable cutoff and macro-bend loss constraints, the maximum effective area that can be achieved is approximately 110 m2 . To further increase the effective area, a depressed cladding layer (low index trench) can be added, as shown in Fig. 6b and c. This suppresses macro-bend losses while keeping the cable cutoff wavelength below 1530 nm. For example, Fig. 7 shows measured bend loss as a function of effective area for design (b) at 1550 nm. For 30 mm bend diameter, the bend loss starts to increase rapidly when effective area is greater than 130 m2 , while for 40 mm bend diameter, the effective area can be as large as 175 m2 and maintain minimal attenuation due to bending.

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Bend Loss (dB/turn)

5 4.5

30 mm bend loss

4

40 mm bendloss

3.5 3 2.5 2 1.5 1 0.5 0 50

75

100

125

150

175

200

Effective Area (mm2)

Fig. 7 Bend loss as a function of effective area for profile design (b) shown in Fig. 6 Table 1 ITU-T macrobend loss fiber specifications G.652.B G.652.D G.654.A G.654.B G.654.C G.654.D G.654.E Macrobend loss at 0.1 0.1 0.5 0.5 0.5 2.0 0.1 1625 nm with 30 mm bend radius (dB/100 turns)

Table 1 shows the ITU-T specifications for maximum macrobend loss for G.652 and G.654-compliant fibers (ITU-T G.652 2016; ITU-T G.654 2016). The specifications are provided at 1625 nm. The maximum cable cutoff wavelength for the G.652 standards is 1260 nm, and these fibers are meant for terrestrial transmission systems which usually involve distances between splices of 2 and 8 km. Splices in terrestrial systems are usually done in splice trays that include a number of fiber turns or loops on each side of the splice. The G.654 standards A-D were designed for submarine fibers with larger effective areas for which the specification for maximum cable cutoff is 1530 nm. A key difference in splicing of submarine fibers compared to terrestrial fibers is that intra-span splices are done in a straight-through configuration (no loops or turns) in which the fiber is re-coated after splicing. The fibers are spliced in something similar to a splice tray configuration only at the repeater sites where it is spliced to fiber jumpers attached to the optical amplifiers. It is worth noting that besides the macrobend loss specifications for submarine fibers as defined by the ITU-T standards, companies that build submarine cables may have their own specifications that a fiber manufacturer must meet. The G.654.E standard recently adopted was also for a larger effective area fiber with cable cutoff of 1530 nm, but since this fiber is nominally intended for terrestrial system deployment, the macrobend loss specification is the same as the G.652 fiber specifications. In addition to macro-bend induced attenuation, attenuation due to micro-bends is also a limiting factor for large effective area optical fibers (Bickham 2010). If the fiber-coating system currently used for standard single-mode fiber (G.652) is

1 Single-Mode Fibers for High Speed and Long-Haul Transmission

15

employed, it has been reported that the fiber effective area may only be increased to approximately 120 m2 due to micro-bend attenuation (Bigot-Astruc et al. 2008). Micro-bending is an attenuation increase caused by high frequency longitudinal perturbations to the waveguide. These perturbations couple power from the guided fundamental mode in the core to higher-order cladding modes that are lost to the outer medium. A phenomenological model introduced by Olshansky captures the importance of treating the glass and coating as a composite system by predicting that the micro-bend losses scale as micro /

a4 E 3=2 b 6 3

(4)

where  micro is attenuation due to micro-bending, a is the core radius, b is the cladding radius,  is the relative refractive index of the core, and E is the elastic modulus of the primary coating layer that surrounds the glass (Olshansky 1975). As mentioned earlier, core delta and core radius are determined by the desired effective area and the cutoff wavelength. These variables are not completely independent and together do not offer a significant path to manage micro-bending sensitivity. This leaves the inner primary modulus as the key factor for addressing increased microbend loss in large effective area fibers. Making the inner primary coating softer helps to cushion the glass from external perturbations and improve the micro-bending performance. The role of primary coating modulus in mitigation of micro-bend induced attenuation has been demonstrated experimentally. In the experiment, fibers of different effective areas were made with two coatings which have higher and lower inner primary moduli (Coating A and B, respectively). Figure 8 shows measured fiber attenuation data for fiber wound under tension on a shipping spool, a condition that promotes microbending. For effective areas between 110 and 115 m2 , the

Attenuation (dB/km)

0.185

0.180 Coating A Coating B 0.175

0.170

0.165

0.160 110-115

120-125

130-135

Effective area (mm2)

Fig. 8 Attenuation measured on shipping reels of fibers with different effective areas and different coatings

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attenuation of fibers with the two coatings are the same. This shows that for fibers with an effective area less than 115 m2 , intrinsic attenuation dominates. For fiber effective areas larger than 120 m2 , the micro-bend loss starts to increase with Coating A, while Coating B minimizes the micro-bend loss to nearly zero for effective areas up to 135 m2 . It is evident that to achieve ultra-low attenuation, a fiber with very large effective area will require a coating with an optimized inner primary modulus to protect against micro-bend induced attenuation. With optimized primary coating, fibers with low loss and effective area of about 150 m2 are now commercially available. The effective area of the fundamental mode can be increased further by increasing the fiber cutoff wavelength beyond the operating wavelength. In this case, the fiber becomes a few mode fiber. A few mode fiber can be used as a quasi-singlemode (QSM) fiber by launching the light into the fundamental mode (Yaman et al. 2010). It has been shown that the effective area of the LP01 mode can be increased to over 200 m2 using this approach (Yaman et al. 2015). The large-effective-area nature of the LP01 mode may help extend the transmission distance considerably. However, an issue of multi-path interference (MPI) arises in QSM transmission that may limit the transmission distance. In general, mode coupling between the LP01 and LP11 modes is expected to be present in practical QSM systems and may induce MPI and therefore signal degradation. MPI must be compensated before the full benefit of the larger effective area of QSM can be realized. It has been shown that MPI can be compensated to a large degree by using digital signal processing techniques in the receiver (Sui et al. 2014).

Quantification of System Level Performance There are many ways to assess and quantify overall system performance of different optical fibers for current high-speed coherent transceivers. Some of these metrics includepbit error rate (BER) or alternatively Q-factor defined as 20 log (Q) where Q D 2erfc1 .2  BER/, a fiber figure of merit (FOM) function, the generalized optical-signal-to-noise ratio (G-OSNR), and the reach length attainable at the forward error correction (FEC) threshold or with a defined Q-factor margin. It is important to mention that there are many factors involved in transmission performance beyond fiber type, including type of optical amplification, span length between optical amplifiers, signal modulation format, symbol rate, FEC overhead, and net coding gain. Here, we will mainly concentrate on the fiber characteristics and attributes that affect modern coherent system performance. We discuss the relative impact of the fiber traits in both conventional long-haul (LH) or ultra-long-haul (ULH) repeatered systems with transmission through a long chain of optical amplifiers, and specialized unrepeatered systems which consist of a long single span with no locally powered amplification provided within the span.

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Long-Haul and Ultra-Long-Haul Transmission Systems We begin by assuming that new coherent transmission systems are designed and deployed without any optical dispersion compensation anywhere in the line system. Thus, all optical amplifiers can be of a single stage configuration and we can assume all dispersion compensation is performed digitally in the digital signal processing (DSP) carried out in the coherent receiver. In principle, some dispersion compensation can also be performed digitally in the transmitter, and somewhat enhanced performance may be obtained in this case. In this context of coherent detection and for standard or Nyquist wavelength division multiplexing (WDM) transmission, a fiber figure of merit (FOM) has been developed as a means of comparing different fibers and their performance (Curri et al. 2013; Hirano et al. 2013; Makovejs et al. 2016a). The FOM is based on the Gaussian noise model of coherent transmission systems (Poggiolini 2012). A simplified version of the FOM is given below in Eq. 5, in which the FOM of a fiber under consideration is defined in a relative sense to a reference fiber.    Aeff n2;ref 2 2 FOM .dB/ D 10 log ˛dB  ˛dB;ref L  3 Aeff;ref n2 3 (5)     Leff 1 1 D C 10 log  10 log 3 Leff;ref 3 Dref In Eq. 5, Aeff is fiber effective area [m2 ], n2 is fiber nonlinear index of refraction [m /W], ˛ dB [dB/km] is fiber attenuation, D is chromatic dispersion [ps/nm/km], L is the span length between amplifiers [km], Leff is the nonlinear effective length .1e˛L /  ˛1 [km], and ’ is the fiber attenuation in linear units [1/km]. Leff D ˛ While some forms of the FOM also include the effect of splice loss to a jumper, we have neglected that here, as we will address splice loss issues later. As it is defined here and based on the Gaussian noise model, the FOM represents the expected difference in 20log(Q) between the fiber under study and the reference fiber in the same system configuration (same link length, span length) if the optimal channel power is used in the system for each fiber. The optimal channel power occurs at the power level for which the noise power from amplified spontaneous emission (ASE) is equal to twice the noise power from nonlinear effects, and represents the channel power that promotes the highest Q-factor and best performance (Poggiolini et al. 2014). As illustrated in Eq. 5, the main fiber parameters that affect overall system performance are attenuation, nonlinear tolerance as governed by effective area and nonlinear index, and chromatic dispersion. As mentioned earlier, attenuation and fiber effective area contribute the most to transmission performance. We therefore begin by temporarily ignoring dispersion effects (essentially assuming equal dispersion for the fiber under study and reference fiber) and examine the FOM as a function of only Aeff and attenuation, as shown in Fig. 9. The reference fiber 2

J. D. Downie et al. FOM (dB)

4-5 3-4 2-3 1-2 0-1 -1-0 0.15

0.155

0.16

0.165

0.17

0.175

0.18

0.185

170 160 150 140 130 120 110 100 90 80 0.19

Effective area (um2)

18

Fiber attenuation (dB/km)

Fig. 9 Fiber FOM as function of attenuation and effective area for 100 km spans in a long-haul repeatered system

for this data has attenuation ˛ D 0.19 dB/km and effective area Aeff D 82 m2 , representative of a generic G.652-compliant standard single-mode fiber. The span length for Fig. 9 is 100 km, typical for many terrestrial LH networks. The data shows clearly that FOM is increased by both reducing fiber attenuation and increasing fiber effective area. The reduction of attenuation from 0.19 to 0.155 dB/km results in about 2.3 dB FOM improvement, while increasing Aeff from 82 to 150 m2 yields almost 1.75 dB advantage for this span length. If the fiber has both 0.155 dB/km attenuation and 150 m2 effective area (as is the case for some state-of-the-art submarine system fibers), the FOM advantage is over 4 dB. The discontinuity between 0.174 and 0.175 dB/km reflects an assumption that attenuation values below 0.175 dB/km are achieved with silica core fibers with lower nonlinear index n2 compared to Ge-doped fibers (Kim et al. 1994; Makovejs et al. 2016a). The n2 values assumed here are 2.3  1020 m2 /W for Ge-doped fiber and 2.1  1020 m2 /W for silica core fiber. The fiber figure of merit function (neglecting dispersion effect) can also be used to understand the relative effects of lower fiber attenuation and greater fiber effective area. The comparison is dependent on the system span length, as given in the second term of Eq. 5. Figure 10 illustrates the equivalent increase in effective area as a function of attenuation reduction. The equivalency is calculated by determining the same increase in FOM with respect to the nominal parameters of the reference fiber described above. This is done for four different span lengths of 120, 100, 80, and 60 km. It is evident that lowering the fiber attenuation has a larger effect for longer spans. For example, a reduction of attenuation (from 0.19 dB/km) of 0.04 dB/km is equivalent to increasing the effective area (above 82 um2 ) by approximately 160, 120, 85, and 60 um2 for span lengths of 120, 100, 80, and 60 km, respectively. To obtain a more comprehensive comparison of different optical fibers in terms of the FOM (including all relevant parameters), we evaluate fiber characteristics that are representative of commercially available optical fibers today. The fibers

1 Single-Mode Fibers for High Speed and Long-Haul Transmission

With respect to reference fiber parameters

180

Equivalent Aeff increase (um2)

19

160

120 km spans

140

100 km spans 80 km spans

120

60 km spans

100 80 60 40 20 0 0

-0.01

-0.02

-0.03

-0.04

-0.05

Attenuation reduction (dB/km)

Fig. 10 Equivalent effective area increase as a function of attenuation reduction for different span lengths, as estimated from FOM Table 2 Realistic optical fiber characteristics ITU standard description

Attenuation (dB/km) Effective area (m2 ) Dispersion at 1550 nm (ps/nm/km) Nonlinear index n2 (m2 /W)

Fiber #1 Conventional G.652

0.19

Fiber #2 Fiber #3 Fiber #4 G.652 with G.652 with G.654.E low loss ultra-low with loss ultra-low loss 0.183 0.162 0.168

Fiber #5 G.654.B with ultra-low loss 0.154

Fiber #6 G.654.D with ultra-low loss 0.154

82

82

82

125

112

150

17

17

17

21

21

21

2.3  1020

2.3  1020 2.1  1020 2.1  1020 2.1  1020 2.1  1020

considered are described in Table 2. Fiber #1 represents a generic G.652 fiber widely used in terrestrial networks. Fiber #2 is a similar fiber with slightly lower attenuation. Fiber #3 is another G.652 fiber made with a silica core that provides significantly lower attenuation. Fiber #4 is a silica core terrestrial fiber with a large effective area compliant with the ITU-T G.654.E standard. Fibers #5 and #6 are silica core submarine system fibers with large effective areas compliant with ITU-T G.654.B and ITU-T G.654.D standards, respectively. The FOM was calculated according to Eq. 5 for Fibers #2–6, relative to the generic G.652 standard single-mode fiber represented by Fiber #1, over a range

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FOM: relative to generic G.652 fiber 6 5

FOM (dB)

4 Fiber #6 Fiber #5

3

Fiber #4 Fiber #3

2

Fiber #2 1 0

60

70

80

90 100 110 Span length (km)

120

130

Fig. 11 FOM as a function of span length for five fibers relative to generic G.652-compliant fiber. Fiber descriptions: Fiber #1: conventional G.652, Fiber #2: G.652 with low loss, Fiber #3: G.652 with ultra-low loss, Fiber #4: G.654.E with ultra-low loss, Fiber #5: G.654.B with ultra-low loss, Fiber #6: G.654.D with ultra-low loss

of span lengths from 60 to 120 km. The results are shown in Fig. 11. As described earlier, the greater impact of lower attenuation with longer span lengths is illustrated by the larger slope of the FOM function with respect to span length for the various silica core fibers. Finally, it is interesting to break out the various components of the FOM functions for each of the fibers at individual span lengths. This data is shown in Fig. 12 for 60 km and 100 km spans. The results confirm that greater benefit is derived from low attenuation with longer spans, while reducing nonlinear effects with larger effective area, lower n2, or greater dispersion has the same impact for any span length. For 60 km spans, the large effective areas of Fibers #4–6 generally provide the leading FOM component, while for 100 km spans, the lower attenuation of those fibers has equal or greater impact on the FOM as effective area.

Raman Gain Considerations While the FOM as constructed in Eq. 5 is independent of the type of optical amplifiers used in a system, it is also worthwhile to examine the nominal expected behavior and performance of the different fibers when used with Raman amplification. Raman amplification is becoming more widely accepted in terrestrial networks because it enables greater reach lengths, especially important for multilevel modulation formats such as PM-16QAM and data rates of 200 Gb/s and higher. In general, the effective noise figure of a distributed Raman amplifier decreases with higher

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21

a 4.5

FOM components (dB)

4 3.5

60 km spans

3 Attenuation

2.5

Effective area

2

n2 Dispersion

1.5 1 0.5 0 Fiber #2

Fiber #3

Fiber #4

Fiber #5

Fiber #6

b 4.5

FOM components (dB)

4

3.5

100 km spans 3 Attenuation

2.5

Effective area 2

n2 Dispersion

1.5

1 0.5 0

Fiber #2

Fiber #3

Fiber #4

Fiber #5

Fiber #6

Fig. 12 Breakout of different FOM components for five fibers: (a) 60 km spans, (b) 100 km spans. Fiber descriptions: Fiber #1: conventional G.652, Fiber #2: G.652 with low loss, Fiber #3: G.652 with ultra-low loss, Fiber #4: G.654.E with ultra-low loss, Fiber #5: G.654.B with ultra-low loss, Fiber #6: G.654.D with ultra-low loss

Raman gain (Evans et al. 2004). Therefore, one aspect of interest with regard to optical fiber is the level of ON/OFF Raman gain achievable with a given level of Raman pump power. For the following analysis, we examine the simplest case of the ideal ON/OFF Raman gain for the six optical fibers considered in the un-depleted pump approximation. In this approximation, the Raman gain can be calculated as

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Gai n.dB/ D

10 gR Ppump Leff ln.10/

(6)

where gR is the fiber’s Raman gain coefficient in units of (1/W/km), Ppump is the Raman pump power in W, and Leff is the effective length of the pump laser at the pump wavelength. The pump effective length is defined as Leff

  1  e ˛p L 1 D  ˛p ˛p

(7)

where ˛ p is the attenuation of the pump wavelength in linear units (1/km). As can be seen from Eq. 6, the Raman gain is affected by fiber characteristics in terms of the pump wavelength attenuation and the Raman gain coefficient gR . The gain coefficient is essentially a function of the fiber material system and the index profile. In particular, since Raman amplification is a nonlinear effect, the gain coefficient is affected by the fiber effective area and the fiber nonlinear index of refraction. To compare the six different fibers in Table 2, we make the simplifying assumption that the gain coefficient of each fiber can be calculated by scaling the gain coefficient of Fiber #1 as gR D gR;#1 

n2 Aeff;#1  n2;#1 Aeff

(8)

By taking the gain coefficient of the generic standard single-mode fiber #1 as about 0.4 W1 km1 , and making a further simple assumption that the attenuation at a pump wavelength of about 1450 nm is 0.05 dB/km higher than the attenuation at 1550 nm for all fibers, it is possible to calculate idealized Raman ON/OFF gain values for the different fibers in an un-depleted pump approximation. The results of that calculation are shown in Fig. 13 for a span length of 100 km and a nominal pump power of 500 mW at 1450 nm. This comparison is simplified, as real systems will likely have more than one pump wavelength to produce relatively flat gain over a wide bandwidth for a large number of optical channels and the un-depleted pump approximation will likely be less accurate. However, this simple analysis is useful for understanding the relative Raman gains likely with the different optical fibers and the same level of pump power. In particular, it is evident that effective area has a significant influence, and fibers with larger effective areas will have smaller ON/OFF Raman gains due to smaller gain coefficients gR . As mentioned earlier, the effective noise figure of a Raman amplifier decreases with increasing Raman gain, so for the same pump power, the overall relative performance advantage of larger Aeff fibers will be slightly smaller than the prediction of FOM in Figs. 11 and 12 in hybrid Raman/EDFA systems in which the Raman gain is less than the total span loss. However, it is also important to realize that fibers with lower attenuation require less Raman gain, which can significantly lessen the pump power demand. An easy way to demonstrate this is to consider systems in which all amplification is obtained through distributed Raman amplifiers. Consider, for example, systems with 100 km spans for which all

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23

Same pump power Nominal ON/OFF Raman gain (dB)

18 16

14 12 10 8 6 4

2 0 Fiber #1

Fiber #2

Fiber #3

Fiber #4

Fiber #5

Fiber #6

Fig. 13 Nominal Raman gain in dB for the same pump power and span length. Fiber descriptions: Fiber #1: conventional G.652, Fiber #2: G.652 with low loss, Fiber #3: G.652 with ultra-low loss, Fiber #4: G.654.E with ultra-low loss, Fiber #5: G.654.B with ultra-low loss, Fiber #6: G.654.D with ultra-low loss

fiber loss is to be compensated with distributed Raman amplification. Then Fiber #1 would require 19 dB of Raman gain (assuming no extra losses) while Fiber #4 would require 16.8 dB of Raman gain. We can use Eq. 6 to calculate relative estimates of the required pump powers needed to produce the gains that would fully compensate for the span losses. The results for all six fibers are shown in Fig. 14, normalized to the pump power required for Fiber #1. We see that even though Fibers #2 and #3 have nearly the same ON/OFF gain for the same pump power as Fiber #1, they require approximately 6% and 17% lower pump power to achieve full span loss compensation, respectively. Similarly, while Fiber #4 has nearly 5.5 dB lower Raman ON/OFF gain than Fiber #1 for the same 500 mW pump power, only about 34% more Raman pump power is required to compensate for the lower span loss of Fiber #4. In that case, under these idealized conditions, Fiber #1 would require 607 mW of pump power and Fiber #4 would require 816 mW of pump power.

Unrepeatered Span Transmission Systems The previous analysis of fiber FOM and the relative effects of attenuation and effective area were in the context of conventional repeatered multispan systems with optical amplifiers compensating for the loss of each span. In contrast, an unrepeatered span system is significantly different and connects a transmitter and receiver pair over a single long span with no active equipment between the terminals. The single span length of these systems can often be up to several hundreds of km

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Pump power to compensate span loss Normalized pump power required

1.6 1.4 1.2 1.0 0.8 0.6 0.4

0.2 0.0 Fiber #1

Fiber #2

Fiber #3

Fiber #4

Fiber #5

Fiber #6

Fig. 14 Normalized pump power required for full span loss compensation with distributed Raman amplification. Fiber descriptions: Fiber #1: conventional G.652, Fiber #2: G.652 with low loss, Fiber #3: G.652 with ultra-low loss, Fiber #4: G.654.E with ultra-low loss, Fiber #5: G.654.B with ultra-low loss, Fiber #6: G.654.D with ultra-low loss

and are designed for both submarine and terrestrial networks. They may connect islands, an island to the mainland, coastal mainland points to each other in a festoon arrangement, or two terrestrial cities with forbidding and difficult terrain between them. In each case, the systems require the transmission of optical signals over a long distance without powered amplifiers between the terminals. A primary goal in the design of many unrepeatered systems is to maximize the reach so the distance between the desired terminal points can be achieved with the desired data rate. There have been many recent research examples of such systems with coherent transmission of 100 and 200 Gb/s channels (Downie et al. 2010; Mongardien et al. 2013). We now examine the relative fiber characteristics that have the most influence on performance and reach in unrepeatered span systems. For repeatered multispan systems, we found that the fiber parameters contributing the most to transmission performance are attenuation and effective area. While dispersion and nonlinear index of refraction have some effect, the range of these parameters is smaller in real fibers and they thus have smaller impact. The same is true for unrepeatered span systems so we will focus again on attenuation and effective area. The fiber FOM given in Eq. 5 was derived in the context of repeatered multispan systems and may not be as directly relevant to unrepeatered span systems. However, we can make relatively simple estimations of the effects of fiber attenuation and effective area on the maximum unrepeatered span length (Downie et al. 2016). To begin, we consider the role of nonlinear tolerance as given by fiber effective area. Similar to a conventional system, the channel launch power into the span will normally be optimized in order to get the best performance. The optimal power

1 Single-Mode Fibers for High Speed and Long-Haul Transmission

25

represents a balance between linear and nonlinear impairments. We recognize that higher optimal channel powers translate into higher optical signal-to-noise (OSNR) values at the receiver. Therefore, the difference in the received OSNR between a fiber under evaluation with effective area Aeff and a reference fiber with effective area Aeff,ref and the same attenuation can be approximated by  OSNR.dB/ D 10 log

Aeff Aeff;ref

 (9)

Under the assumption that a given OSNR value is required at the receiver to produce the desired Q-factor value, the difference in attainable span length L between the evaluated and reference fibers can be simply related to the common attenuation of the fibers as L.km/ D

OSNR.dB/ ˛dB

(10)

The results predicted by Eqs. 9 and 10 for a hypothetical reference fiber with effective area Aeff, ref D 80 m2 are shown in Fig. 15 for two different values of fiber attenuation. The actual predicted increase in reach L is relatively small and less than 22 km, even for fiber effective areas as large as 170 m2 with very low attenuation of 0.15 dB/km. This simple analysis does not account for lower Raman gain that accompanies larger fiber effective area, which may serve to reduce the OSNR and thus further reduce the reach increase compared to the reference fiber.

Increase in reach (km)

25

20

15

10 0.150 dB/km

5 0.170 dB/km 0

80

100

120 140 Fiber effective area (mm2)

160

180

Fig. 15 Increase in the reach or span length of unrepeatered span systems as a function of fiber effective area relative to 80 m2

26

J. D. Downie et al.

We next consider the effect of lower fiber attenuation on the reach of an unrepeatered span system. There are many different designs for these systems, including the use of forward and backward pumped Raman amplification, remote optically pumped amplifiers (ROPAs) in mid-span locations, higher-order Raman pumping, etc. (Chang et al. 2015; Huang et al. 2017). However, for a given type of system design, a commonly used means of quantifying the performance of the design is by the maximum span loss that can be tolerated for a given level of signal quality and capacity. To compare the reach L of a fiber with attenuation ˛ dB to the reach Lref of a reference fiber with attenuation ˛ dB, ref , we can equate the maximum loss accommodated by the system design as Loss max .dB/ D ˛dB  L D ˛dB;ref  Lref

(11)

It is then straightforward to calculate the difference in reach length L D L  Lref between the fiber under evaluation and the reference fiber as   ˛dB;ref  ˛dB L.km/ D  Loss max .dB/ (12) ˛dB ˛dB;ref Advanced long reach unrepeatered spans may have maximum loss values between 60 and 100 dB (Chang et al. 2015; Huang et al. 2017). The data in Fig. 16 represents the increase in reach length L in km for fibers with attenuation values 0. This filter has two poles that are reciprocals of each other: r ˙ z1 ; z2 D 1 C 2 

r

1C

r 2 1 2

(73)

with z1 inside the unit circle mapping to a causal sequence and z2 outside the unit circle mapping to an anti-causal sequence. The inverse z-transform of Eq. 73 is:

w

;k

D

˛ r jkj ˛ 1  ˛2

(74)

q 2  1 C 2r  1. The filter in Eq. 74 is a two-sided decaying where ˛ D 1 C 2r  exponential symmetric about the origin. The exponential decay arises due to phase estimates at times k  l far from the symbol k of interest being progressively less accurate estimators of  k , so they are given decreasing emphasis in computing Ok . The symmetry of the filter is due to the accuracy of the estimator k  l being dependent only on jlj. In the limit of low phase noise compared with AWGN (2 n20 ), the decay rate of the filter is slow (˛ ! 1) to exploit the longer temporal coherence of  k . Conversely, in the limit of high phase noise compared with AWGN (2  n20 ), the filter has P a fast decay rate as k-l rapidly becomes inaccurate estimators of  k . Finally, 1 kD1 w ;k D 1, so Eq. 74 is an unbiased estimator. Since it is impractical to implement a two-sided exponential with infinite tails, Eq. 74 can be approximated by an infinite impulse response (IIR) filter in parallel with a finite impulse response (FIR) filter as shown in Fig. 31. The performance of the MMSE estimator has been studied in Ip and Kahn (2007b).

Fig. 31 Emulating the ideal two-sided exponential filter with an IIR filter in parallel with an FIR filter

4 Optical Coherent Detection and Digital Signal Processing of Channel Impairments

189

Phase Estimation in the Presence of Data Modulation While the carrier phase estimation algorithm in section “Phase Estimation in the Absence of Data Modulation” is sufficient for a system which transmits a pilot tone alongside the signal, most systems in operation today do not use any pilot tones. The phase measurement k at symbol k must therefore be extracted from the modulated signal. Like linear equalization, this can be achieved by the use of training symbols (decision-aided), symbol decisions (decision-directed), or by exploiting known properties of the modulated signal (nondecision-aided). Decision-aided phase estimation is not common, as the carrier phase will be unknown to the receiver between training symbols. Since laser phase evolves much faster than the channel impulse response, large numbers of training symbols will have to be inserted at regular intervals, reducing the rate of useful data throughput. Nondecision-aided (NDA) and decision-aided (DD) phase estimation can be accomplished using the feedforward structure shown in Fig. 32a. In the upper path, an instantaneous phase estimate k is first obtained from the signal. k and is then passed through a filter W (with delay ) to produce an MMSE estimate Ok of the carrier phase. This phase is then used to de-rotate the signal in the bottom path, which has been appropriately delayed. Compared with feedforward carrier estimation described for a pilot tone in section “Phase Estimation in the Absence of Data Modulation,” the only additional component is the instantaneous phase estimator. NDA phase estimation is commonly used for M-ary PSK, as the M-fold rotational symmetry of the constellation means that raising the signal to be the M-th power removes data modulation. This method of phase estimation is also commonly referred to as the Viterbi-Viterbi algorithm (Viterbi and Viterbi 1983). Let the set of possible transmitted symbols be xm D ej2m/M , m 2 [0, M  1]. Raising the received signal to the M-th power yields: M  ykM D xk e j k C nk D e jM k C mk

(75)

  M P M  j k M p p xk e nk pD1 p is a summation of signal-noise beat terms contributing a noise vector to the phasor e jM k . The instantaneous phase can be found from the angle of ykM and divided by 1/M (Fig. 32b): where e jM k is the desired term dependent on  k , and mk D

k

D

 1 arg ykM  k C n0k M

(76)

where M n0k is the angular projection of mk . At high SNR ( s ), it can be shown that n0k is approximately i.i.d. Gaussian with zero mean and variance (Ip and Kahn 2007b):

190

E. Ip

a

Instantaneous Phase Estimator

Symbol Detection

b Phase Unwrap

c

Decision Phase Unwrap

Fig. 32 (a) Feedforward carrier phase estimation and compensation. The instantaneous phase estimator can be (b) nondecision aided (NDA) or (c) decision-directed (DD)

n20 D .M; s /

1 s

(77)

2 M pC1 where .M;  / D pŠs . p For non-(D)PSK constellations, k can be estimated using a DD instantaneous phase estimator shown in Fig. 32c. A major difference between the NDA and the DD structures is that DD requires an initial estimate of the carrier phase Qk . In the DD algorithm, the received signal yk is first de-rotated by Qk . Provided Qk is not significantly different than the real carrier phase  k , the de-rotated constellation will be properly aligned and can be detected by a decision device: 1 PM pD1 2M 2



i h Q xO k D yk e j k

D

(78)

4 Optical Coherent Detection and Digital Signal Processing of Channel Impairments

191

At high SNR, the decision is correct (xO k D xk ) with high probability. This allows the instantaneous carrier phase to be estimated: k

D arg .yk /  arg .xO k / D k C n0k

(79)

where n0k is the angular projection of noise nk in the direction orthogonal to the rotated signal xk e j k . It can be shown that at high SNR, n0k are approximately i.i.d. and Gaussian with zero mean and variance: N2 0 D i h h Where D 12 E jxj2 E

1 jxj2

i

1 s

(80)

is equal to half times a “constellation penalty.”

Phase Unwrapping The instantaneous phase produced the NDA and DD phase estimators, as well as the pilot tone phase estimator covered in section “Phase Estimation in the Absence of Data Modulation,” which need to be phase unwrapped before passing through the MMSE filter W . This is a consequence of the arg() function output being limited to – and , as angles differing by integer multiples of 2 are indistinguishable. In the absence of phase unwrapping, the NDA phase estimator output is constrained between /M and /M, whereas the DD output is constrained between – and . Since  k is a Wiener process, its value is unconstrained. Phase unwrapping has been studied in Taylor (2004) and involves adding integer multiples of 2/M (or 2) to the output of the NDA (or DD) phase estimator to ensure that the magnitude of the phase difference between adjacent symbols is less than /M (or ). Phase unwrapping can cause “cycle slipping” and is a highly nonlinear phenomenon (Meyr et al. 1997).

Combining Laser Phase Noise Compensation with Linear Equalization In presence of both laser phase noise and an LTI channel, a system that uses receiver-side impairment compensation has the model shown in Fig. 33. The phase of the transmitter and receiver local oscillator lasers is denoted as  tx (t) and  rx (t), respectively. The receiver can invert the impairments in reverse order as shown by the operations in the dotted box in Fig. 33. For clarity, the function blocks in the channel inverter are shown as continuous-time functions. Provided the receiver signal is sampled above the Nyquist criterion (section “Sampling Rate Requirement”), the equivalent operations can be performed in DSP. Since multiplication by laser phase noise is non-commutable with dispersion, the DSP blocks cannot be reordered without loss of performance. Moreover, as  tx (t),  rx (t), and h(t) are time-varying, the DSP compensators Otx .t /, Orx .t /, and

192

E. Ip

Channel

Transmitter Laser

Receiver LO Laser

Receiver-side Impairment Compensation

Fig. 33 Receiver-side impairment compensation taking into account laser phase noise and an LTI channel

w(t) must be adaptive. Previously, we saw in linear equalization in section “Linear Equalization” that adaptation requires the receiver to feedback the error between the output of each DSP block and the desired output. This is also true for laser phase noise compensation. For the Tx phase noise compensator, the desired output is the transmitted symbols. Since the linear equalizer precedes the Tx phase compensator, O the desired output is dk e j tx;k , where dk is given in Table 1 for the various O adaptive linear equalizer algorithms. The error signal is therefore ©k e j tx;k . By extension, the error for each block is obtained by backpropagating (The principle of backpropagating the error to adapt a nonlinear system has been studied in the field of neural networks (Haykin 1998). The usage of backpropagation here should not be confused with the nonlinear compensation technique of “backward propagation” introduced in section “Nonlinear Compensation”) the error ©k through all the other blocks downstream from it, as shown in Fig. 34a. It is observed that the Rx phase noise compensator is a digital LO controlled by a phase-locked loop (PLL), where the feedback delay is equal to the sum of the latencies of all the other DSP blocks, including the linear equalizers in the impairment compensation and error feedback paths. This feedback delay is functionally analogous to the physical delay of a hardware PLL. Delay reduces linewidth tolerance. It was shown in Barry and Kahn (1992) that for large feedback delays  , the linewidth requirement scales as 2  . Most of the delay in  comes from the group delays of w and w1 , as these have to be causal filters. The linewidth requirement for the Rx local oscillator laser will therefore scale as 2 rx Ncd Ts , where Ncd is the memory length of the channel due to CD given in Eq. 50. By contrast, the linewidth requirement for the Tx laser scales as 2 tx Ts , as the feedback path does not enclose the filters w and w1 . Hence for a receiver that uses feedback adaptation only, as shown in Fig. 34a, the linewidth requirement on the receiver LO is much more severe than the Tx laser. Laser phase noise tolerance can be increased by using feedforward adaptation. Consider Fig. 34b, in which the operations in Fig. 34a are labeled “Iteration #1” and where the error feedback path is labeled as the “parameter update” circuit. A second copy of the received signal is delayed by the feedback latency. The receiver uses the parameters obtained in the initial adaptation (Iteration #1) to process the received signal a second time (Iteration #2) to produce a better estimate of the transmitted

4 Optical Coherent Detection and Digital Signal Processing of Channel Impairments Fig. 34 Compensation of phase noise and LTI impairments using (a) feedback adaptation only and (b) using both feedback and feedforward adaptation

193

a

Σ Compute Error

εk Parameter Update

b

Iteration #1

Compute Error

εk Feedback Feedforward

Delay Iteration #2

Parameter Update

signal. This additional processing represents “feedforward” operation. It is possible to increase the number of iterations of impairment compensation C error estimation C parameter update to obtain asymptotically better performance.

Nonlinear Compensation Digital Backpropagation Signal propagation in fiber is ultimately distorted by a combination of both chromatic dispersion and Kerr nonlinearity, as described by the NLSE introduced in section “The Fiber Channel.” A typical characteristic of transmission performance versus signal power is shown in Fig. 35. The vertical axis denotes signal quality (QSNR ), which we define as signal power divided by the total variance of “interference” arising from noise and nonlinearity. At low launch power, optical fiber is well modeled as an LTI channel, so a 1 dB increase in launch power results in a 1 dB improvement in signal quality. As signal power is increased, the nonlinear term in the NLSE will begin to dominate.

194 Fig. 35 Signal quality versus power characteristic with and without nonlinear compensation

E. Ip

Linear Regime

Nonlinear Regime Nonlinear Compensation

Linear Equalization

Power For fibers having “large” CD (say, larger than 2 ps/nm/km) and where in-line CD compensation is not used, it has been shown that nonlinear distortion is well approximated as a Gaussian noise (Poggiolini 2012; Poggiolini et al. 2014; Mecozzi and Essiambre 2012; Savory 2013), whose variance grows as the cube of signal power. The cubic scaling is due to the Kerr effect being a third-order nonlinearity. Assuming nonlinearity is not compensated, QSNR will have the form: QSNR D

P N C P 3

(81)

where P is signal power, N is total noise power added by the link, and is a parameter that depends on the fibers propagated through. From Eq. 81, it is observed that QSNR will decrease by 2 dB for every 1 dB increase in launch power at high powers. Traditionally, nonlinear impairments in DWDM systems are categorized into the following types (Agrawal 2002): 1. Self-phase modulation (SPM): Nonlinearity arising in a channel as a result of its own intensity 2. Cross-phase modulation (XPM): Nonlinearity arising in a channel as a result of the intensity of other channels in a WDM system 3. Four-wave mixing (FWM): Nonlinearity arising from three interacting fields at different wavelengths producing a nonlinear field at a fourth wavelength within the bandwidth of the channel of interest. 4. Nonlinear phase noise (NLPN): Nonlinear interaction between signal and noise (Gordon and Mollenauer 1990). The first three types of nonlinear impairments are deterministic given the full electric field of the WDM signal. In the absence of noise, these nonlinearities can be completely compensated as the NLSE is an invertible equation. It is therefore possible to simultaneously compensate CD and nonlinearity by solving the inverse NLSE:   @E O CN O E D D @z

(82)

4 Optical Coherent Detection and Digital Signal Processing of Channel Impairments

195

O and N O are the linear and nonlinear operators defined in Eq. 20. The where D operation performed by Eq. 82 is analogous to passing the received signal E(L, t) through a fictitious fiber with opposite signs of loss, dispersion, and nonlinearity, yielding an estimate of the transmitted signal E(0, t). This method of nonlinear compensation by solving the inverse NLSE is commonly referred to as “backward propagation” or “backpropagation” (BP). When implemented in DSP, it is known as digital backpropagation (DBP). In the context of Eq. 82, linear equalization of CD and PMD, whether performed optically using DCF or digitally using a linear equalizer, can be viewed as a “simplified DBP” taking only into account the linear operator. Even mid-span phase conjugation can be viewed as a form of BP, as inverting the signal phase at midpoint causes the second half of the transmission medium to become a backpropagation on the first half of the link (Chowdhury and Essiambre 2004). DBP was first proposed as a transmitter-side electronic pre-compensation algorithm in Essiambre et al. (2006) (Fig. 36a), since before the advent of a dualpolarization coherent receiver, the complex-valued electric field is only available at the transmitter. The transmitter’s DSP solves the inverse NLSE to find the modulator drive signal necessary so that the received optical waveform is free of both linear and nonlinear distortions after propagation. When the dual-polarization coherent receiver became available, receiver-side DBP was made possible. Receiver-side DBP was studied in Mateo et al. (2008) and Ip and Kahn (2008) (Fig. 36b). It is also possible to split the DBP operation between the transmitter and receiver (Fig. 36c). Since noise added by in-line amplifiers causes the actual waveform in forward transmission to be different to that digitally backpropagated, splitting the DBP operation 50/50 between the transmitter and receiver yields the best performance. When nonlinear compensation is employed, the DSP operations performed at the transmitter and receiver are shown in Fig. 37. Since BP is only concerned with the electric field, it is a universal algorithm that operates independently of the modulation format. Moreover, as DBP simultaneously compensates both LTI impairments and Kerr nonlinearity, the only additional DSP operation required at the receiver is laser phase noise compensation. In practice, DBP can only take into account the linear impairments known to the receiver (and/or) transmitter. Polarization rotation and PMD are generally unknown. Therefore, a short linear equalizer after DBP is usually required prior to carrier recovery.

Split-Step Fourier Method In DBP, the inverse NLSE in Eq. 82 is normally solved using the split-step Fourier method (SSFM) (Agrawal 2001), which divides the fiber channel into sections (Fig. 38). To backpropagate the signal through a section of fiber from zCh to z, it is possible to use a non-iterative, asymmetric SSFM (NA-SSFM) (Fig. 39a):     O .z C h/ exp hD O E .z C h; t / E .z; t /  exp hN

(83)

196

a

E. Ip

b

Tx-side Backpropagation , , ,

c

, ,

,

,

, ,

Tx-side Backpropagation , , ,

, ,

,

,

Rx-side Backpropagation , , ,

,

Rx-side Backpropagation , , ,

,

TRANSMITTER

Carrier Recovery

Linear Equalizer

Backward Prop.

ADC

Fiber Channel

O-E Downconvert

E-O Upconvert

DAC

Data In

Backward Prop.

Fig. 36 Digital backpropagation performed at the (a) transmitter side, (b) receiver side, and (c) splitting between transmitter and receiver sides

Data Out

RECEIVER

Fig. 37 Nonlinear compensation transmitter and receiver

a

... Fiber Section

b

Fiber Section

Fiber Section

... Fiber Section

Fiber Section

Fiber Section

Fig. 38 (a) Forward and (b) backward propagation for a fiber channel modeled as a concatenation of Nsec fiber sections

4 Optical Coherent Detection and Digital Signal Processing of Channel Impairments

197

a

Iteration

b

Fig. 39 Backpropagation through one section of fiber using the (a) non-iterative asymmetric SSFM and (b) iterative symmetric SSFM

Alternatively, a more accurate iterative, symmetric SSFM (IS-SSFM) can also be used (Fig. 39b): !    O .z C h/ C N.z/ O hO N hO E .z; t /  exp  D exp h exp  D E .z C h; t / 2 2 2 (84) 

The latter algorithm is “symmetric” because the linear operator is split equally on each side of the integral of the nonlinear operator. In Eq. 84, this integral has O been approximated using the trapezoidal rule (Agrawal 2001). Since N.z/ depends on E(z, t), whose value is initially unknown, an iterative algorithm is required to solve Eq. 84. In the NA-SSFM, the need for iteration is removed by approximating O the integral of N.z/ using a one-sided rectangle. The IS-SSFM is more accurate than the NA-SSFM but is more computationally expensive. The linear operator is most efficiently computed in the frequency domain as an all-pass filter with polynomial phase: n    o  n o  O E z0 ; t D exp h F D O F exp hD E z0 ; ! D e ’h=2 e j .ˇ2 !

2 =2ŠCˇ ! 3 =3Š 3

/h R . ; / e j !. =2/¢ 1 RH .; / E z0 ; ! (85)

where R( , ) was defined in Eq. 22. The nonlinear operator is most easily evaluated in the time-domain as a phase shift proportional to the instantaneous signal power in both polarizations:         O E z0 ; t D exp j 8  kE z0 ; t k2 h E z0 ; t exp hN 9

(86)

Solving the SSFM requires the use of FFTs and IFFTs to switch between frequency and time, which accounts for the algorithm’s high computational cost.

198

E. Ip

Step-Size Requirement To compute the NLSE accurately, the step size h is normally chosen so that the phase accumulated by the linear and nonlinear step (in time or frequency) is less than :

h  min

8
0), the soliton amplitude decreases (increases), while its width increases (decreases), i.e., the soliton broadens (is compressed) along the fiber. Let us now return to the full problem and study the effect of the perturbation (23) on the dynamics of bright solitons. Following the same procedure, i.e., substituting Eq. (23) into Eqs. (29), (30), (31), (32) and performing the integrations, we find that the soliton parameters evolve according to the following system: d 2 D .3 C 2ı2 /; dz 3

(34)

8 d D  R 4 ; dz 15

(35)

dt0 1 D  C .3ˇ  3  2/ C 3ˇ 2 2 ; dz 3

(36)

  1 8 d D  .  3ˇ/2 C ˇ.2  2/q 2 C .2   2 /  R t0 4 : dz 2 15

(37)

Although our findings in this case are more complicated, it is still possible to arrive at simple analytical results. Indeed, first we observe that Eq. (34) can be solved analytically to provide the functional form of .z/, which is found to be: 2 .z/ D

3K e4z ; 1  2Kıe4z

KD

2 .0/ : 3 C 2ı2 .0/

(38)

Then, the frequency .z/ can be obtained from Eq. (35) by simply integrating the above expression for . Finally, having found .z/ and .z/, integration of Eqs. (36) and (37) yield, respectively, the functional forms of t0 .z/ and .z/. In Fig. 1 we show a typical evolution for a bright soliton under Eq. (22) with  D ı D ˇ D  D  D R D 0:01. In this figure, the top panel depicts the complete evolution of the soliton and its center (also drawn in a white line the result

280

T. P. Horikis and D. J. Frantzeskakis

Fig. 1 Top panel: The contour plot of the evolution of a unit amplitude bright soliton (i.e., .0/ D 1) under Eq. (22) with  D ı D ˇ D  D  D R D 0:01. The white dashed line is the soliton’s center t0 as predicted by the perturbation theory. Bottom panel: The evolution of the amplitude corresponding to the top panel and the prediction of the theory

of the perturbation theory), while in the bottom panel, we show a direct comparison between the numerically found amplitude and the theoretically predicted from the perturbation theory.

Perturbation Theory for Dark Solitons Next, we consider the case of the normal dispersion regime (s D C1), i.e., the case of dark solitons. As we will see, the fact that the dark soliton is a more complicated object than the bright one – since it is composed by both the background and the pulse on top of it – makes the problem more difficult. For this reason, we first present a general perturbation theory for dark solitons and, in the next section, we will apply the derived results to Eq. (8).

The Background To begin our considerations, we first rewrite Eq. (22) in a slightly different form, using capital letter U for the unknown electric field envelope (the reason will become obvious shorty), namely: 1 iUz  Ut t C jU j2 U D "F ŒU : 2

(39)

6 Perturbations of Solitons in Optical Fibers

281

Further, as is relevant in this case, we will assume nonvanishing boundary conditions at infinity, i.e., jU j 6! 0 as t ! ˙1. The effect the perturbation has on the behavior of the solution at infinity is independent of any local phenomena, such as localized pulses which do not decay at infinity, i.e., dark solitons. In the case of a cw background, which is relevant to perturbation problems with dark solitons, we have Ut t ! 0 as t ! ˙1, and the evolution of the background at either end, U ! U ˙ .z/, is given by the equation:

i

d ˙ U C jU ˙ j2 U ˙ D "F ŒU ˙ : dz

(40)

˙

We write U ˙ .z/ D u˙ .z/ei .z/ , where u˙ .z/ > 0 and ˙ .z/ are both real functions of z. Then, the imaginary and real parts of Eq. (40), respectively, read: i h d ˙ ˙ ˙ u D "Im F Œu˙ ei ei ; dz i h d ˙ ˙ ˙ D .u˙ /2  "Re F Œu˙ ei ei =u˙ : dz

(41a) (41b)

The above equations completely describe the adiabatic evolution of the background under the influence of the perturbation F Œu. Although this is true for all choices of perturbation, we will further restrict ourselves to perturbations which maintain the phase symmetry of Eq. (39); i.e., F ŒU .z; t /ei  D F ŒU .z; t /ei . As we show next, this is a sufficient condition to keep the magnitude of the background equal on either side and a property of most commonly considered perturbations, including the one in Eq. (23). We now assume that at z D 0; uC .0/ D u .0/; then, since u˙ .z/ satisfy the same equation, the evolution is the same for all z. Hence, uC .z/ D u .z/  u1 .z/. While this restriction is convenient, the essentials of the method presented here apply in general. The equations for the background evolution (41) can now be further reduced by considering the phase difference  1 .z/ D C .z/   .z/ which is the parameter related to the depth of a dark soliton (see below); here ˙ .z/ represents the phase as t ! ˙1, respectively: d u1 D "Im ŒF Œu1  ; dz d  1 D 0: dz

(42a) (42b)

Thus, while the magnitude of the background evolves adiabatically, the phase difference remains unaffected by the perturbation.

282

T. P. Horikis and D. J. Frantzeskakis

The Soliton and the Shelf Let us now focus on the evolution of a dark soliton under perturbation. To simplify the calculations, introduce the transformation: Z z  2 U D u exp u1 .s/ ds ; (43) 0

so Eq. (39) becomes: 1 iuz  ut t C .juj2  u21 /u D "F Œu: 2

(44)

It is now convenient to express the dark soliton solution of the unperturbed equation in a slightly different form than the one in Eq. (14), namely: us .t; z/ D fA C iBtanh ŒB .t  Az  t0 /g exp .i0 / ;

(45)

where the parameters of the soliton “core” A; B; t0 , and 0 , denoting the velocity, amplitude, center, and phase of the soliton, are all real. Furthermore, the background amplitude is .A2 C B 2 /1=2 D u1 , and the phase difference across the soliton is 2 tan1 .B=A/, with A ¤ 0. When the soliton velocity A D 0, Eq. (45) describes a black soliton, which has a phase difference of , as was also explained above. Below, we employ the method of multiple scales by introducing a slow scale variable Z D "z, with the parameters A, B, t0 , and 0 being functions of Z. In addition, we seek the solution of Eq. (44) in the form of the following perturbation expansion: u D u0 .Z; z; t / C "u1 .Z; z; t / C O."2 /

(46)

The order O.1/ approximation u0 .Z; z; t / should satisfy the slowly varying boundaries from Eqs. (42), which means that two of the parameters are already pinned down, namely, A.Z/ D u1 .Z/ cos. 1 =2/ and B.Z/ D u1 .Z/ sin. 1 =2/, and we take 0 .0/ D 0. Now we consider the general case of a dark soliton with velocity A.Z/; also recall A2 .Z/ C B 2 .Z/ D u21 .Z/. Let u D q exp.i /, where q > 0Rand are real z functions of z and t , and introduce moving frame of reference T Dt  0 A."s/ds t0 and  D z. This way, Eq. (44), with u D u.; T; Z/, becomes: 1 iu  iAuT  uT T C .juj2  u21 /u D "F Œu 2

(47)

Then, using u D q exp.i /, we obtain: i.q C i  q/  iA.qT C i T q/ 

 1 qT T C i2 T qT C .i T T  T2 /q 2

C q 3  u21 q D "F Œq; :

(48)

6 Perturbations of Solitons in Optical Fibers

283

The imaginary and real parts of the above equation, respectively, read: 1 q D AqT C .2 T qT C q T T / C "Im ŒF Œq;  ; 2 1  q D A T q  .qT T  T2 q/ C .jqj2  u21 /q  "Re ŒF Œq;  : 2

(49a) (49b)

We now write Eq. (49b) in terms of the slow evolution variable  D "Z and series expansions q D q0 C"q1 CO."2 / and D 0 C" 1 CO."2 /. At O.1/ the equations are satisfied by the soliton solution (45). On the other hand, at O."/ we have: q1 D Aq1T C

1 Œ2. 0T q1T C 1T q0T / C 1T T q0 C 0T T q1  C Im ŒF Œu0   q0Z ; 2 (50)

1 2 1 q0 D  0 q1 C A. 0T q1 C 1T q0 /  .q1T T  0T q1  2 0T q0 1T / 2 C 3q02 q1  u21 q1  Re ŒF Œu0   0Z q0 ;

(51)

where u0 D q0 exp.i 0 /. We look for stationary solutions at O."/: Aq1T C

1 f2. 0T q1T C 1T q0T / C 1T T q0 C 0T T q1 g C Im ŒF Œu0   q0Z D 0; 2 (52a)

1 2 q1  2 0T q0 1T / C 3q02 q1  u21 q1 A. 0T q1 C 1T q0 /  .q1T T  0T 2  Re ŒF Œu0   0Z q0 D 0; (52b) where q0Z 0Z

  1 BZ 1 2 AAZ C BBZ tanh .x/ q0 C q0T  t0Z ; D 2 B  BZ  t0Z C 0Z : D .ABZ  BAZ / tanh.x/q02 C 0T B

(53a) (53b)

Consider, now, Eq. (52a) in the limit T ! ˙1; using q0 ! u1 and u1Z D ImF Œu1 , we obtain: ˙ C Aq1T

u1 ˙ D 0: 2 1T T

(54)

We assume that q1 tends to a constant with respect to t , i.e., q1T ! 0 as t ! ˙1. As a result, 1T T ! 0. Then q1 and 1T both tend asymptotically to constants as

284

T. P. Horikis and D. J. Frantzeskakis

t ! ˙1, which corresponds to a shelf developing around the soliton (Ablowitz et al. 2011a). Substituting 0T into Eq. (52b), and in the limit T ! ˙1, we get: ˙ A 1T C 2u1 q1˙ D Re ŒF Œu1  =u1 ˙

.ABZ  BAZ / C 0Z u21

(55)

We define  0 by:  0 D 2 tan

1



B ; A

(56a)

the phase change across the core soliton, as in the unperturbed case. This is consistent with the soliton parameters A and B being expressed in terms of background magnitude, u1 , and phase change,  0 :  A D u1 cos

 0 2



 ;

B D u1 sin

 0 2

;

(56b)

again as in the unperturbed case. Using Z˙ D Re ŒF Œu1  =u1 and substituting Eq. (56b) into Eq. (55), we find: ˙ C 2u1 q1˙ D Z˙ ˙ A 1T

 0Z C 0Z : 2

(57)

Adiabatic Dynamics Next, following a procedure reminiscent to the one used for the bright solitons, we will now employ the renormalized conservation equations to determine the slow evolution of the soliton parameters A and 0Z , as well as the shelf parameters q1˙ and 1t˙ . The evolution of the renormalized integrals of the dark soliton read:  Z 1 d 2 dHds D" E u C 2Re F ŒuNuz dt ; dz dZ 1 1 Z 1 dEds D 2"Im F Œu1 u1  F ŒuNudt; dz 1 Z 1 dPds D 2"Re F ŒuNut dt dz 1 Z 1 dRds D P C 2"Im t .F Œu1 u1  F ŒuNu/ dt: dz 1

(58a) (58b) (58c) (58d)

However, more work is required in order to find t0 . Note that if we find A, then B D .u21  A2 /1=2 . The edge of the shelf still propagates with velocity V .Z/ D u1 .Z/;

6 Perturbations of Solitons in Optical Fibers

285

however, the speed may now vary in z. In terms of the moving frame of reference, the boundaries of the shelf are: Z



SL ./ D 

Œu1 ."s/ C A."s/ ds;

(59a)

Œu1 ."s/  A."s/ ds;

(59b)

0

Z



SR ./ D C 0

where SL and SR denote the positions (in T ) of the left and right boundaries of the shelf, respectively, at . Note that A  u1 for all Z; thus, the soliton cannot overtake the shelf. The inner region consists of the soliton core and the shelf expanding around it, while the outer region consists of the infinite boundary conditions characterized by Eqs. (42). We begin with the evolution equation for the Hamiltonian (58a), which reads: d d

Z

1 1



 Z 1    2  1 1 jut j2 C .u21  juj2 /2 dt D " u21 Z u1  juj2 dt 2 2 1 Z 1 F ŒuNu dt: (60) C 2"Re 1

Substituting u D .q0 C "q1 / expŒi. 0 C " 1 / and changing variables to the moving frame of reference, we have up to O."/: d d

Z

1 1

 2    2 .q0T C 0T q02 / C .u21  q02 /2 dT D 2" u21 Z Z

Z

1 1

 2  u1  q02 dT

1

 4"Re

F Œu0 ANu0T dT;

(61)

1

where, both here and later on, u0 D q0 exp.i 0 /. The Hamiltonian is unique among the evolution Eqs. (58) in that the contribution of the shelf appears only at O."2 / or higher, and to O."/ may be ignored. We now put in the soliton form to get: 2B 2 BZ D .u21 /Z B  ARe

Z

1

F Œu0 Nu0T dT:

(62)

1

Taking a derivative with respect to Z of the equation u21 D A2 C B 2 , we get: .u21 /Z D 2AAZ C 2BBZ ;

(63)

which can be used to consolidate Eq. (62) to the form: Z

1

2BAZ D Re

F Œu0 Nu0T dT: 1

(64)

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T. P. Horikis and D. J. Frantzeskakis

The evolution equations for energy (58b) and momentum (58c) both remain the same after transforming to the moving frame of reference: d d

Z

1 1

 2  u1  juj2 dT D 2"Im

d Im d

Z

Z Z

1

1

(65)

F ŒuNuT dT:

(66)

1

uNuT dT D 2"Re 1

.F Œu1 u1  F ŒuNu/ dT; 1

1

The inner region over which q1 and 1 are relevant is T 2 ŒSL ./; SR ./, and outside this region q1 D 1T D 0. At O.1/ the equations are obviously satisfied, while at O."/ we have: BZ 

d d

Z

Z

SR ./

1

.F Œu1 u1  F Œu0 Nu0 / dT;

q0 q1 d T D Im

d 2.AB/Z  d

(67a)

1

SL ./

Z

SR ./

SL ./

  2 0T q0 q1 C 1T q02 dT D 2Re

Z

1

F Œu0 Nu0T dT: 1

(67b) Since the integrands on the left-hand side are not functions of , we can apply the fundamental theorem of calculus to arrive at:   BZ  u1 .u1  A/q1C C .u1 C A/q1 D Im

Z

1

.F Œu1 u1  F Œu0 Nu0 / dT; 1

(68a)   C  D 2Re C .u1 C A/ 1T 2.AB/Z C u21 .u1  A/ 1T

Z

1

F Œu0 Nu0T dT: 1

(68b) We are left now with the evolution of the center of energy: d d

Z

1 1

t .u21  juj2 /dt D Im

Z

Z

1

1

t .F Œu1 u1  F ŒuNu/ dt;

uNut dt C 2"Im 1

1

(69)

which, after transforming to the moving frame of reference, becomes: d d Z

Z

1

(T C

1 1

uNuT dT C 2"Im 1

 0

Z

1

D Im

Z

1

A C t0 ).u21  juj2 /dt

(T C

Z



A C t0 ) .F Œu1 u1  F ŒuNu/ dT:

0

(70)

6 Perturbations of Solitons in Optical Fibers

287

After rearranging some terms, we have: d d

Z

1 1

Z

T .u21  juj2 /dT !



C

A C t0 0

d d

Z

(71a)

1

1

 u21  juj2 dT  "2Im

Z



1

.F Œu1 u1  F ŒuNu/ dT 1

(71b) Z

1



CA 1

Z

 u21  juj2 dT C Im 1

D "t0Z 1



Z

1

uNuT dT

(71c)

1

 u21  juj2 dT C 2"Im

Z

1

T .F Œu1 u1  F ŒuNu/ dT:

(71d)

1

The terms on line (71a) yield: d d

Z

1 1

  T .u21  juj2 /dT D 2 SR .u1  A/q1C C SL .u1 C A/q1 u1 :

(72)

The terms on line (71b) reproduce the energy equation (65) and cancel out. The terms on line (71c) are calculated up to O."/ using the previous results by integrating the energy and momentum equation (68):   E./ D 2B  2 SR .Z/q1C  SL .Z/q1 u1 C "E1 .Z/ C O."2 /;   I ./ D 2AB  u21 SR .Z/ 1tC  SR .Z/ 1t C "I1 .Z/ C O."2 /;

(73) (74)

where it is noticed that d =d  D "d =dZ, while SR and SL are O.1="/ in terms of Z. Now, putting everything together in terms of slow evolution variable Z D " we get from (71): Z

1

"2Bt0Z D 2"Im 1

  T F Œu1 u1  F Œu0 u0 d T " C AE1 .Z/ C "I1 .Z/

    C f2u1 SR .u1  A/q1C C SL .u1 C A/q1 C 2u1 A SR q1C  SL q1   (75) C u21 SR 1tC  SL 1t g: After some cancelations, this breaks into O.1/ terms:     C  D 0;  SL 1T 2 SR q1C C SL q1 C SR 1T

(76)

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T. P. Horikis and D. J. Frantzeskakis

and O."/ terms which include t0Z , while higher-order energy and momentum terms have not been determined. The six Eqs. (57), (64), (68a), (68b), and (76) can now be ˙ used to solve for the set of six parameters q1˙ , 1t˙ .D 1T /, A, and 0 :

Z

1

2BAz D Re 1

Z

 F Œus .us /t dt ; 

1

Bt0z D Im

t .F Œu1 u1  1

Z

(77)

1

u1 0z D Im

.F Œu1 u1  1

F Œus us /

F Œus us /

dt ;

(78)

 dt C Re fF Œu1 g ;

(79)

BBz D u1 u1z  AAz

(80)

u21  0z D 2ABz  2BAz

(81)

q1˙ D

1 0z ˙  0z ; 2 u1 A

1t˙ D 2q1˙ :

(82)

The above equations may now be solved from top to bottom.

Dark Solitons Under Perturbations Our analysis starts with the dynamics of the soliton background. Assuming that u.z; t ! 1/ D u0 .z/, we derive from Eq. (6) the equation: iu0z  ju0 j2 u0 D i u0 C i ıju0 j2 u0 :

(83)

Then, employing the polar decomposition u0 D u1 .z/ exp.i .z//, we obtain: u01 D . C ıu21 /u1 ;

(84)

and 0 D u21 , where primes denote differentiation with respect to z. Notice that the role of the term of strength ı is now more obvious: a nontrivial equilibrium (constant solution) exists if  ı < 0, which is u21 D =ı. We focus here on these solutions, i.e., solutions that tend to stabilize the soliton, by keeping its parameters constant. Employing the results of the previous section, we find that the rest of soliton parameters evolve according to the following equations: A0 D

4 2 8 R A4 C ıA3  R u21 A2 15 3 15  4 ı 2 C  C u1 A C R u41 ; 3 15

(85)

6 Perturbations of Solitons in Optical Fibers

289

  2 4 2 A  2ˇ C 2 C u21 ; D 2ˇ    3 3  Bz 2B  3 C 4u21 ı C 2ıA2 :   00 D u1 3u1 t00

(86) (87)

These equations show that while the evolution of the soliton center, described by the equation T00 D A C t00 , is affected by all parameters of Eq. (6) [directly or indirectly from A.z/, the solution parameters, i.e., the background, soliton core, and shelf, only depend on  , ı, and R . Thus, soliton stabilization can be targeted accordingly. We now proceed by presenting numerical results for Eq. (6), which will be compared to the above analytical predictions. We use the parameter values ˇ D  D  D 0:01 and vary  , ı, and R in order to investigate stabilization of the background and the soliton parameters. A typical result of the simulations, for  D 0:025, ı D 0:02, and R D 0:02, with initial condition a unit-amplitude black soliton, is shown in Fig. 2. In this figure, top panel depicts the evolution of the soliton andRthe emergence of the shelf, propagating with a speed u1 .z/, so z that its edge is 0 u1 .s/ds. Additionally, bottom panel shows the evolution of the background amplitude and soliton velocity. Let us now return to the system of Eqs. (84) and (85). Stable fixed points of this system correspond to stable solitons travelling on top of a constant background with a constant speed. Below, we identify two such solitons, namely, a gray and 30 25

0.8

z

20

0.6

15

0.4

10

0.2

5 −20

0 t

20

1 soliton parameters

Fig. 2 Top panel: contour plot showing the evolution of a unit-amplitude black soliton; analytical predictions are depicted by the solid (white) lines, for the propagation of the shelf’s edge, and the dotted (white) line, for the evolution of the soliton center. Bottom panel: evolution of u1 .z/ and A.z/; solid lines (circles/squares) correspond to analytical (numerical) results

u∞(z)

0.8 0.6 0.4 0.2 0

A(z) 0

10

z

20

30

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T. P. Horikis and D. J. Frantzeskakis

a black one, supported in the presence (R ¤ 0) and in the absence (R D 0) of the SRS effect, respectively. In both cases, the background assumes the same form: u21 D =ı [cf. Eq. (84)] for  ı < 0, i.e., for linear loss and nonlinear gain, or vice versa. We start with the case R ¤ 0. Substituting u21 D =ı in Eq. (85) and seeking stationary solutions for the soliton velocity, we arrive at a fourth-order algebraic equation for A. We find that there exists only one root of this equation, which does not violate the fundamental relationship A2 C B 2 D u21 ; this root is: A D .5ı 2 C q 25ı 4  16 ıR2 /=.4ıR /. Thus, a stable soliton exists for:

u21

 D ; ı

5ı 2 C AD

q 25ı 4  16 ıR2 4ıR

:

(88)

Note that since  ı < 0, the quantity under the square root is always positive. Taking  D ı D R D 0:02, the soliton propagation for the above initial data is shown in the top panel of Fig. 3. Evidently, the soliton is characterized by a stable evolution (the background is fixed, and the center moves with constant speed), while suppression of the shelf is also observed. In fact, the calculated shelf is found to be of order O.103 /, i.e., an order of magnitude less that that of the perturbation.

30

Fig. 3 The panels are similar to those in Fig. 2, but now for the stable gray soliton, characterized by the parameters given in Eq. (88)

0.8

25

0.7

z

20 15

0.6

10

0.5

5

0.4 −20

20

0 t

soliton parameters

1.2 u∞(z)

1 0.8 0.6

A(z)

0.4 0.2

0

10

z

20

30

6 Perturbations of Solitons in Optical Fibers

291

The bottom panel of Fig. 3 also illustrates the background amplitude u1 and soliton velocity A.z/. Next, we consider the situation where the SRS effect is absent, i.e., R D 0. In this case, Eqs. (84) and (85) lead to the following equations for the background and soliton velocity:  u21 D  ; ı

A0 D

2 .ıA2   /A; 3

(89)

Obviously, the above equation for the velocity depicts a stationary solution A D 0 (recall that  ı < 0) that corresponds to a black soliton. Hence, when SRS is absent (which would result in a frequency downshift causing the soliton to move), a stable black soliton can exist, as seen in Fig. 4. The evolution of u1 .z/ and A.z/ follows a qualitatively similar behavior to that observed in the bottom panel of Fig. 3 and, hence, is not shown. Here, it is important to mention that while these cases for u1 and A stabilize the soliton, this does not mean that the shelf is no longer present. The shelf is always present in the solution of the perturbed NLS equation, even if not noticeable due to its magnitude in the numerical simulations. In any case, it does not affect the single soliton propagation, but it will affect soliton interactions; nevertheless, addressing this problem is beyond the scope of this work. Notice, also, suppression of the shelf will result in the destabilization of the soliton. We also briefly consider the case where gain/loss terms are absent, i.e.,  D ı D 0. In this particular case, the dark soliton dynamics is merely driven by the SRS effect. Indeed, now the evolution of the background and soliton velocity is described by the following equations: u01 D 0;

A0 D

4 R .A2  u21 /2 : 15

(90)

Since A2 ¤ u21 , the right-hand side of the second equation is always positive and, thus, the minimum of the soliton density is always ascending and will finally 30

0.8

25 0.6

20 z

Fig. 4 Similar to the top panel of Fig. 2, but for the stable black soliton (for R D 0), characterized by the parameters given in Eq. (89), for  D ı D 0:02

15

0.4

10

0.2

5 −20

0 t

20

292

T. P. Horikis and D. J. Frantzeskakis

be merged to the stationary background. Thus, obviously, no stable pulse (in the sense of stationary parameters) exists. We note that (90) recover the results obtained in Uzunov and Gerdjikov (1993) and Kivshar and Yang (1994), while numerical simulations illustrating the soliton evolution in this case provided results similar to those reported in Kivshar and Yang (1994) (and, hence, not repeated here). Finally, it is important to make the following comment. The focus here was on the dynamics stemming from stationary nontrivial solutions of Eq. (84). However, this equation has the following general solution: u21 .z/ D

 u20 e 2z ;  C ıu20  ıu20 e 2z

(91)

where u0 D u1 .0/. This suggests that there is a (finite) distance, z? , for which the background exhibits blowup. Indeed, the denominator is zero when:   1 z? D ln 1 C 2 : 2 ıu0

(92)

The unexpected feature here, which was avoided above by choosing j j D ıju0 j2 , is that the addition of the term ıjuj2 that counters the effects of the linear term may result in a blowup of the background in finite z, even when the other soliton parameters remain finite. Also, Eq. (91) shows that an equilibrium can also be reached in finite z when the denominator is a multiple of the numerator. While this will stabilize the background, it does not mean necessarily that it stabilizes the other parameters.

Beyond the Adiabatic Theory: Soliton Radiation In the previous sections, we employed the adiabatic approximation to study perturbations of either bright or dark solitons in optical fibers. We have shown that when perturbations – such as damping, higher-order dispersion, higher-order nonlinearity, etc. – come into play, solitons do not propagate as undistorted objects but, instead, their physical properties (shape, velocity, center, etc.) over the propagation distance. Furthermore, during soliton evolution, energy radiation can additionally be excited, which can affect the soliton dynamics in nontrivial ways. To motivate a short discussion on the effect of radiation that we append below, it is relevant to mention at this point the following. Soliton propagation in non-ideal optical fibers has attracted much attention, particularly in view of the relevance to high band-pass optical communication systems. In that regard, it should be recalled that there is a strong link between phase fluctuation and changes in the soliton velocity, resulting in jitter in the pulse train. One important source of phase fluctuation is background radiation present in the fiber, which accompanies the soliton pulses. This radiation may result either from perturbing influences in the

6 Perturbations of Solitons in Optical Fibers

293

fiber (such as the ones studied in the previous sections) or be present as part of the input pulse injected into the fiber (i.e., in the case where the initial condition is a perturbed soliton). To be more specific, let us consider the anomalous dispersion regime and rewrite the perturbed NLS, Eq. (22), in the following form: iut C suxx  2juj2 u D iF;

(93)

where, as before, s D 1 (s D C1) corresponds to the anomalous (normal) dispersion regime. The perturbation iF in the NLS modifies the soliton solution in two distinct ways: (i) The soliton parameters, which were constants of the motion in the unperturbed case, now vary with distance down the fiber. If the perturbation is small, this change is adiabatic and can be studied in the framework of the adiabatic perturbation theory exposed in the previous sections. (ii) The perturbation is responsible also for the generation of a background radiation field, which is superimposed on the soliton pulse. Depending on the nature of the perturbation, this can exhibit quite complicated resonance features (see, e.g., Wai et al. 1986; Elgin 1992). To incorporate the effect of radiation in the context of the perturbed NLS Eq. (93), one may assume that the field u.x; t / departs only a little from being a pure soliton; then, one may seek solutions of Eq. (93) of the form (see, e.g., Kivshar and Malomed 1989): u.z; t / D us .z; t / C ıu.z; t /;

(94)

where us .z; t / has the functional form of the (bright or dark) soliton solution [cf. Eqs. (12) and (14)], but with the soliton parameters depending on the propagation distance – as in the previous sections – and ıu.z; t /, with ıu.0; t / D 0, is the radiation emitted by the soliton. Assume that the amplitude of the perturbing radiation field ıu is much smaller than unity and its energy is much smaller than that of the soliton. Substituting Eq. (94) into the NLS and keeping first-order terms, one obtains a linear propagation equation for the radiation field, namely: iıuz C sıut t C 4jus j2 ıu C 2u2s ıu D iF;

(95)

where, to simplify our notation below, star is used to denote complex conjugate. Here, even in the simpler case where a soliton evolves in an ideal fiber (i.e., F D 0), a difficulty arises when an attempt is made to solve the above equation using standard techniques. For instance, if a Fourier transformation with respect to the t variable is used, the presence of us results in convolution terms which prove intractable to further analysis.

294

T. P. Horikis and D. J. Frantzeskakis

The expression u D us C ıu is apparently not unique. Small changes in the parameters of us can be offset by modifications to ıu, leaving the total u unchanged. However, consider a field that has this form at some t . If this field is permitted to propagate sufficiently far, some unique soliton will emerge, along with a small dispersed field of the order of ıu that has left the vicinity of the soliton. This unique soliton, extrapolated back to the location of the perturbation, is called the emergent soliton (Gordon 1992). The emergent soliton is the same as us only if ıu is purely dispersive; in other words, radiation is actually a dispersive field. Two cases are usually considered: the first case considers an ideal fiber in the sense that subsequent pulse evolution is described by the NLS equation, with the pulse input to the fiber consisting of soliton plus radiation; in the second case, the fiber is not ideal, and pulse evolution is described by a perturbed form of the NLS. The study of some of the properties of the radiation field, either accompanying the soliton pulse or generated by it, relies on inverse scattering theory (Ablowitz and Segur 1981). In the case of the anomalous dispersion regime, i.e., for bright solitons, particular solutions to a scattering problem associated with the latter – the Jost function solutions – are combined bilinearly to produce a complete basis of continuum states, onto which the radiation field is projected; this process resembles the situation when a simple function, say x.t /, is projected onto the kernel exp.i!t / in standard Fourier analysis. The manner in which the radiation field is projected onto these states and, more importantly, the reconstruction of this radiation field from a knowledge of these continuum modes together with a minimal set of “scattering data” is briefly outlined below. In the case of dispersive perturbations, one may follow Gordon (1992); Elgin (1993) and introduce the associate field formalism. This approach provides results consistent with the ones obtained from a perturbation expansion for the ZakharovShabat inverse scattering transform (Kaup 1978b) (see also Haus 1997). According to the associate field formalism, the required form for ıu.z; t / reads: ıu.z; t / D ft t C 2 ft   2 f C u2s f  ;

(96)

where  D 2tanh.2t /, while f evolves according to

ifz D ft t 

i 4

Z

C1

1

exp.2i t / 2 C 2

Z

C1

.F  1 N 1 C F 2 N 2 / dt d ;

(97)

1

where i , N i , , and N i are the components of Jost functions associated with the direct scattering problem of the NLS (Ablowitz and Segur 1981). Obviously, when F D 0, the associate field evolves according to: ifz D ft t :

(98)

6 Perturbations of Solitons in Optical Fibers

295

It then follows that the radiation field ıq is again described by Eq. (96); however, in this case ıu.0; t / is a nonzero function of t , else ıu.z; t /  0 due to the homogeneous character of Eq. (98). Let us now briefly discuss the case of the normal dispersion regime, i.e., radiation for dark solitons. In this case, the situation is different. The scattering problem for the NLS equation with normal dispersion is different, and the closure of the relative Jost functions has not been shown. In addition, because these Jost functions differ from the focusing case, the direct analogue of the associate field has not been found, to date. Nevertheless, in some cases, e.g., in the case of the defocusing NLS perturbed by the third-order dispersion, it is possible to show the following (Karpman 1993; Afanasjev et al. 1996). Under the action of the thirdorder dispersion, radiation in the form of an oscillating tail is formed from the right of the soliton, and its front propagates with group velocity vg , which is different from the dark soliton velocity v and the speed of sound c. Performing an asymptotic analysis, it is possible to show that, for sufficiently small coefficient ˇ of the thirdorder dispersion term, the amplitude A of the tail depends on ˇ exponentially for fixed v, namely (Karpman 1993; Afanasjev et al. 1996): " A  exp 



#

; p u1 ˇ 1  v 2 =c 2

(99)

where u1 is the background amplitude. This dependence of A on ˇ is similar to that for bright solitons, where it was found that the tail amplitude is exponentially small in the third-order dispersion coefficient, namely, A  exp.1=ˇ/, and can be calculated by asymptotic analysis “beyond all orders” (Wai et al. 1986). These results are also consistent with the soliton robustness hypothesis (Menyuk 1993). p Notice that the presence of the factor 1  v 2 =c 2 in the exponent in Eq. (99) shows that the radiation amplitude becomes exponentially small for any fixed ˇ in the limit v ! c, i.e., for small-amplitude dark solitons (Kivshar and Afanasjev 1991; Frantzeskakis 1996). It is also important to note that radiation also affects the interactions between solitons, a fact that is quite important for applications both in optical communications and in fiber lasers. For instance, an important effect that was predicted is that some perturbations tend to attenuate the interaction between bright solitons and may even lead to formation of bound states (BSs). This way, weakly overlapping solitons in the dissipatively perturbed NLS equation form a set of BSs, as was predicted in theory (Malomed 1991, 1992) and observed in numerical experiments (Agrawal 1991). In Malomed (1993), it was demonstrated that additional dissipative terms, e.g., a term describing the stimulated Raman scattering in a nonlinear optical fiber, destroy all the BSs provided the corresponding coefficient exceeds a certain critical value. On the other hand, in the absence of dissipation, and in the framework of the NLS equation with the third-order dispersion, it was demonstrated that two solitons – or a whole array of solitons – may form BSs, interacting with each other via emitted radiation (Malomed 1993). It is important to note that tightly BS solitons

296

T. P. Horikis and D. J. Frantzeskakis

have been experimentally observed in nonlinear polarization rotation, figure-ofeight, carbon nanotubes, and graphene-based mode-locked fiber lasers (Akhmediev et al. 1998; Grelu et al. 2002; Seong and Kim 2002; Tang et al. 2005; Zhao et al.; Wu et al. 2011; Li et al. 2012; Gui et al. 2013; Tsatourian et al. 2013). It is also relevant to mention that the interactions between dark solitons in the presence of perturbations have also been studied but less extensively. For instance, in Afanasjev et al. (1998), a perturbed defocusing NLS equation with a nonlinear saturable gain – actually a complex quintic Ginzburg-Landau equation – was studied. The focus of this work was on the formation and stability of two darksoliton BSs, and it was found that such BSs do exist and appear to be fully stable in the quintic equation. A related dissipative equation – the so-called power-energy saturating model – was analyzed in Ablowitz et al. (2011b, 2013), and the effect of perturbations in the interactions between dark solitons was numerically studied. Notice that theoretical results reported in Afanasjev et al. (1998) and Ablowitz et al. (2011b, 2013) were in quantitative agreement with experimental and numerical results of the recent work (Guo et al. 2016), where the controlled generation of bright or dark solitons in a fiber laser was studied.

Summary and Conclusions In this chapter, we have reviewed the dynamics of bright and dark solitons in optical fibers. Our exposition started with the introduction of the nonlinear Schrödinger (NLS) equation, which models soliton propagation in the ideal case (perfect fiber). Soliton solutions of the latter were introduced, and their fundamental properties were briefly discussed. We then focused on more realistic situations, where the NLS was modified to include extra terms accounting for various effects arising in real optical fibers and optical fiber links. These effects include loss, gain (for the compensation of loss), as well as third-order dispersion, self-steepening, and stimulated Raman scattering, which are particularly relevant for sub-picosecond pulses. We have thus focused on the problem of studying the evolution of solitons under these perturbations. To analytically deal with this problem, we have adopted the rather general framework of the adiabatic perturbation theory. According to this approach, the functional form of the solitons does not change, but their parameters (e.g., amplitude, width, velocity, etc.) change along the fiber – i.e., they become unknown functions of the propagation distance. Our analysis was performed independently for the cases of the bright and dark solitons. In the case of bright solitons, where the NLS model is supplemented with vanishing boundary conditions at infinity, the problem was rather straightforward to study. Indeed, using the evolution of the conserved quantities, it was possible to derive a set of ordinary differential equations (ODEs) for the soliton parameters that is even solvable analytically. In fact, once the evolution of the bright soliton

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amplitude was determined, all other parameters were found by simple integrations. The numerical results were shown to be in excellent agreement with the analytical predictions. The study of the dynamics of dark solitons under perturbations proved to be more demanding. The problems in this case arise from the fact that the dark soliton is a more complicated object than its bright counterpart, because it is composed by a continuous-wave (cw) background and the soliton core. Thus, in this case, it was necessary to consider separately the dynamics of the dark soliton’s constituents. In the case of the background, we have found that dissipative perturbations do have a nontrivial effect: boundary conditions in the vicinity of the soliton core and at infinity become unequal, which suggests the emergence of a “shelf,” i.e., a linear wave that develops around the soliton core, and thus bridging the relevant difference. To account for this effect, it was necessary to develop a multiscale moving boundary layer theory, which accurately captures the dynamics of the soliton background, core, and shelf. We have also briefly discussed effects that cannot be captured by the adiabatic perturbation theory. Such an effect is the perturbation-induced emission of radiation, which is particularly important for applications in optical communications and in the context of fiber lasers. We have thus presented a general framework and reviewed some analytical results for the case of bright solitons; pertinent results for dark solitons are not available, to the best of our knowledge, so far. Nevertheless, we also presented results for dark solitons, which can be obtained, in some cases (e.g., the NLS perturbed by the third-order dispersion), by means of proper asymptotic analysis. We also briefly described the effect of radiation on soliton interactions. We have thus explained that some perturbations tend to attenuate the interaction between bright solitons and may lead to the formation of soliton bound states; this effect was also observed in experiments with fiber lasers.

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6 Perturbations of Solitons in Optical Fibers P.K.A. Wai, C.R. Menyuk, Y.C. Lee, H.H. Chen, Opt. Lett. 11, 464 (1986) X. Wu et al., Opt. Commun. 284, 3615 (2011) V.E. Zakharov, A.B. Shabat, Sov. Phys. JETP. 34, 62 (1972) V.E. Zakharov, A.B. Shabat, Sov. Phys. JETP. 37, 823 (1973) V.E. Zakharov, E.A. Kuznetsov, Physica D 18, 455 (1986) B. Zhao et al., Phys. Rev. E 70, 067602 (2004)

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Emission of Dispersive Waves from Solitons in Axially Varying Optical Fibers A. Kudlinski, A. Mussot, Matteo Conforti, and D. V. Skryabin

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emission of a Dispersive Wave from a Soliton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental Soliton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dispersive Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generation of Dispersive Waves from Solitons in Axially Varying Optical Fibers . . . . . . . . . Axially Varying Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emission of Multiple Dispersive Waves Along the Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . Cascading of Dispersive Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transformation of a Dispersive Wave into a Fundamental Soliton . . . . . . . . . . . . . . . . . . . Emission of Polychromatic Dispersive Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generation of a Dispersive Wave Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The possibility of tailoring the guidance properties of optical fibers along the same direction as the evolution of the optical field allows to explore new directions in nonlinear fiber optics. The new degree of freedom offered by axially varying optical fibers enables to revisit well-established nonlinear phenomena

A. Kudlinski ()· A. Mussot · M. Conforti CNRS, UMR 8523 – PhLAM – Physique des Lasers Atomes et Molécules, University of Lille, Lille, France e-mail: [email protected]; [email protected]; [email protected] D. V. Skryabin Department of Nanophotonics and Metamaterials, ITMO University, St Petersburg, Russia Department of Physics, University of Bath, Bath, UK e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 G.-D. Peng (ed.), Handbook of Optical Fibers, https://doi.org/10.1007/978-981-10-7087-7_10

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and even to discover novel short pulse nonlinear dynamics. Here we study the impact of meter-scale longitudinal variations of group-velocity dispersion on the propagation of bright solitons and on their associated dispersive waves. We show that the longitudinal tailoring of fiber properties allows to observe experimentally unique dispersive wave dynamics, such as the emission of cascaded, multiple, or polychromatic dispersive waves.

Introduction Since its discovery in the frame of the Korteweg-de Vries equation by Zabusky and Kruskal (1965) and Gardner et al. (1967), the concept of solitons has been extended to many other systems described by integrable equations (Ablowitz et al. 1973), including the nonlinear Schrödinger equation (NLSE) (Zakharov and Shabat 1972) widely used to study nonlinear pulse propagation in optical fibers (Mollenauer et al. 1980; Hasegawa and Matsumoto 2002; Dudley et al. 2006). However, in real-world fibers, intrinsic higher-order dispersive and nonlinear effects break the integrability of the NLSE and therefore perturb the invariant propagation of solitons (Kivshar and Malomed 1989). Fundamental solitons, on the contrary to higher-order ones, are very stable in optical fibers, and they usually survive these perturbations, by continuously adapting their temporal and spectral shapes (Elgin 1993). The robustness of fundamental solitons has been exploited in the context of dispersionmanaged optical communications in which the periodic evolution of losses and gain due to fiber attenuation and amplifiers is compensated by a periodic arrangement of dispersion and/or nonlinearity (Turitsyn et al. 2012). The so-called dispersionmanaged solitons are propagating over kilometer-long systems, and they are usually relatively broad (in the picosecond duration scale), making them weakly altered by higher-order dispersion. In fact, in some cases, third-order dispersion might even help to stabilize them (Hizanidis et al. 1998). In the case of much narrower solitons (in the hundreds of femtosecond time scale), higher-order dispersion plays a much more significant role. In the case of a perturbation due to thirdorder dispersion, for example, a short soliton propagating near the zero-dispersion wavelength (ZDW) loses energy into a dispersive wave (also called resonant or Cherenkov radiation) across the ZDW (Wai et al. 1986, 1987) and experiences a spectral recoil in the opposite spectral direction in order to conserve the overall energy (Akhmediev and Karlsson 1995). This very well-known process has been studied extensively from theoretical and experimental points of views (Wai et al. 1986, 1987; Akhmediev and Karlsson 1995; Cristiani et al. 2004; Erkintalo et al. 2012; Webb et al. 2013; Conforti and Trillo 2013), in particular in the context of supercontinuum generation in which it plays a crucial role in the early dynamics (Dudley et al. 2006; Skryabin and Gorbach 2010). Following their emission, dispersive waves can also collide with solitons in the presence of Raman effect, leading to their nonlinear interaction (Yulin et al. 2004; Efimov et al. 2005; Skryabin and Yulin 2005). Understanding this nonlinear wave mixing process has been a key in extending supercontinuum sources toward the blue/ultraviolet spectral region

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(Skryabin and Gorbach 2010; Gorbach et al. 2006; Gorbach and Skryabin 2007) and also toward long wavelengths in some specific cases (Chapman et al. 2010). In this chapter, we study the process of dispersive wave emission from a soliton in various axially varying optical fibers. We show that the longitudinal variation of guiding properties allows to observe experimentally new and unique dynamics regarding dispersive waves. The first section introduces the basics of dispersive wave emission from a fundamental soliton, and the second one focuses on the peculiarities of this process in axially varying fibers.

Emission of a Dispersive Wave from a Soliton In this section, we will introduce the basic concepts of fundamental temporal soliton propagation in optical fiber with second-order dispersion (or group-velocity dispersion (GVD)) and Kerr nonlinearity, as well as radiation of dispersive wave in the presence of third-order dispersion.

Fundamental Soliton The nonlinear propagation of light in dispersive and nonlinear optical fibers is described by the nonlinear Schrödinger equation (NLSE) ˇ2 @2 A @A D i C i  jAj2 A @z 2 @2 t

(1)

Here, t is the retarded time in the frame traveling at the group velocity vg of the input pulse. A.z; t / is the envelope of the electric field, ˇ2 is the fiber GVD coefficient, and  is the Kerr nonlinear parameter defined using the standard definition from Agrawal (2012). The fundamental soliton is an analytical solution of Eq. 1 (Zakharov and Shabat 1972) when the second-order dispersion coefficient is negative (anomalous GVD) and the nonlinear parameter  is positive. Its amplitude takes the form A.z; t / D

  p t P0 sech T0

(2)

where P0 is the soliton peak power and T0 its duration. The soliton is termed fundamental when the following condition is fulfilled: P0 D

jˇ2 j  T02

(3)

Figure 1 shows the results of a numerical integration of Eq. 1 in a standard telecommunication single-mode fiber. The second-order dispersion coefficient is ˇ2 D 6:05  1028 s2 /m and the nonlinear parameter is  D 2:08 W1 . km1 .

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Fig. 1 Simulation of soliton propagation using Eq. 1, i.e., neglecting third-order dispersion. (a) Input (black dashed line) and output (red solid line) spectra in a standard telecommunication single-mode fiber of 200 m length. (b) Spectral dynamics along propagation, with the same colorscale as in (d). (c) Corresponding input (black dashed line) and output (red solid line) temporal profiles. (d) Temporal dynamics along propagation. Simulation parameters are given in the text

The input pulse is a fundamental soliton centered at 1310 nm with a duration T0 of 100 fs and a peak power of 29.08 W (in accordance with Eq. 3). Figure 1a–d shows that the input pulse propagates without any spectral or temporal modification along the fiber, respectively. This is the most striking feature of temporal fundamental solitons, and this is the main reason why they have attracted so much interest both from fundamental and applicative point of views (Hasegawa and Matsumoto 2002; Taylor et al. 1996; Mollenauer and Smith 1988).

Dispersive Wave In practice, however, one cannot always neglect the fiber third-order dispersion. In particular, it plays a major role when the soliton is located near the zero-dispersion wavelength (ZDW) of the fiber (Wai et al. 1986, 1987), i.e., when the soliton spectrum overlaps the normal GVD region located across the ZDW. In this case, Eq. 1 has to be rewritten in order to take into account the fiber third-order dispersion. It takes the form

7 Emission of Dispersive Waves from Solitons in Axially Varying Optical Fibers

ˇ2 @2 A ˇ3 @3 A @A D i C C i  jAj2 A @z 2 @2 t 6 @3 t

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(4)

where ˇ3 is the third-order dispersion coefficient of the fiber. The third-order dispersion drastically modifies the soliton dynamics as compared to the case in which it is neglected. This is illustrated in Fig. 2 where we have simulated the propagation of the same input pulse as above using Eq. 4. We have considered a ˇ3 value of 6:9  1041 s3 /m, the other fiber parameters being the same ones as above. The fiber ZDW (represented by the black dotted line in Fig 2a, b) is 1302 nm. The output spectrum obtained after 200 m (red solid line in Fig. 2a) strongly differs from the input one (black dashed line). First, it exhibits a strong peak around 1280 nm corresponding to a dispersive wave. Second, the soliton spectrum is distorted, and its central wavelength has redshifted from 1310 to 1313 nm. This is the spectral recoil accompanying the emission of the dispersive wave. The spectral dynamics displayed in Fig. 2b shows that the spectrum initially broadens toward the shortwavelength region and that the dispersive wave starts to form at about 50 m of propagation. In the time domain (Fig. 2c), it appears at the trailing edge of the soliton (red solid line) as a broad and relatively distorted pulse (see inset). Since the soliton has experienced a spectral recoil, it has slightly decelerated as compared to the fiber input. It has also lost some energy, which has been radiated into the dispersive wave. This can be further observed in the temporal dynamics plot of Fig. 2d, where we can see the slow deceleration of the soliton as well as the continuous radiation of a broad dispersive wave along propagation. The frequency at which the dispersive wave is emitted can be predicted from phase-matching arguments between the two radiations as (Akhmediev and Karlsson 1995) ˇ2 2 ˇ3 3  P0 ˝ C ˝ D 2 6 2

(5)

where ˝ D !S  !DW is the frequency separation between the soliton at !S and the dispersive wave at !DW . Solving Eq. 5 with the parameters of our study gives a dispersive wave wavelength of 1283 nm, in excellent agreement with the numerical simulation results of Fig 2a, b. The process of dispersive wave emission from a soliton is very well-known and explained as long as optical fibers are uniform in length: when the soliton spectrum crosses the ZDW, the part of soliton energy located in the normal GVD region is radiated into a dispersive wave. This causes a spectral recoil of the soliton which thus moves away from the ZDW so that the emission process becomes less and less efficient and cannot occur again. Consequently, there is a unique dispersive wave which is generated and appears as a sharp spectral peak whose frequency does not change with propagation, as observed from Fig. 2b. However, the process can be radically different in optical fibers which are not uniform as a function of length, i.e., in axially varying fibers. This will be the focus of the remaining of this chapter.

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Fig. 2 Simulation of soliton propagation using Eq. 4, i.e., taking third-order dispersion into account. (a) Input (black dashed line) and output (red solid line) spectra in a standard telecommunication single-mode fiber of 200 m length. (b) Spectral dynamics along propagation, with the same colorscale as in (d). (c) Corresponding input (black dashed line) and output (red solid line) temporal profiles. (d) Temporal dynamics along propagation. Simulation parameters are given in the text. The black dotted lines in (a) and (b) represent the fiber ZDW

Generation of Dispersive Waves from Solitons in Axially Varying Optical Fibers Axially Varying Optical Fibers For the fabrication of axially varying optical fibers, the longitudinal evolution of the fiber diameter is controlled by adjusting the evolution of drawing speed with time (which is related to fiber length) using a servo-control system. This process is ruled by the conservation of glass mass between the preform and the fiber: s dFiber D dPreform

VPreform VFiber

(6)

where dPreform, Fiber are, respectively, the preform and fiber outer diameter and VPreform, Fiber are, respectively, the preform feed into the furnace and the drawing

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Fig. 3 (a) Evolution of the outer diameter as a function of fiber length for three different axially varying fibers measured during the fiber fabrication process. (b) Corresponding calculated ZDW. For the second fiber (red case), the ZDW plotted in (b) corresponds to the second one (located at long wavelengths)

capstan speed. In our process, dPreform and VPreform are fixed, and VFiber is adjusted with a desired f .z/ function (where z is the longitudinal space coordinate along the fiber), which results in a modulation of the fiber outer diameter dFiber .z/ and thus of the overall fiber structure. This results in a modulation of the mode(s) propagation constant(s) and thus of all guiding properties. Figure 3a shows three examples of axially varying fibers used hereafter. These curves show the evolution of the outer diameter recorded during the fiber drawing process. Figure 3b shows the corresponding calculated ZDW. The example represented in red corresponds to a fiber with two ZDWs, and the plot in Fig. 3b corresponds to the second ZDW. In the range of diameter variations investigated here, the ZDW approximately follows a linear dependence with fiber diameter.

Emission of Multiple Dispersive Waves Along the Fiber As recalled above, in uniform fibers, the soliton experiences a redshift (spectral recoil) during the generation of a dispersive wave. This redshift is usually strongly reinforced by the Raman-induced soliton self-frequency shift. As a consequence, the soliton moves far away from the ZDW, so that the dispersive wave emission process cannot occur again. We will first show an example in which the use of a suitable axially varying fiber allows for a single soliton to emit multiple dispersive waves along the fiber. Indeed, this can occur if the fiber ZDW moves along propagation so that it “hits” the soliton during its redshift following the emission of a dispersive wave (Arteaga-Sierra et al. 2014; Billet et al. 2014). In this case, the overlap between the soliton spectrum and the normal GVD region can become large enough to initiate the emission of a new dispersive wave. This is illustrated with numerical simulations and experiments in Fig. 4a, b, respectively. For these numerical simulations, the stimulated Raman scattering term has been added to

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Fig. 4 (a) Numerical simulations and (b) experimental results in the spectral domain versus fiber length showing the emission of three distinct dispersive waves from a single soliton in an axially varying fiber. The evolution of the ZDW with fiber length is represented by black solid lines

Eq. 4, which now writes   Z @A ˇ2 @2 A ˇ3 @3 A 0 0 2 0 /jA.t  t /j dt D i C C i  A R.t @z 2 @2 t 6 @3 t

(7)

where R.t / D .1  fR /ı.t / C fR hR .t / includes both Kerr and Raman effects, where hR .t / corresponds to the Raman response function (fR D 0:18) taken from Hollenbeck and Cantrell (2002). We consider a fiber with a ZDW which evolves along the fiber (blue curve in Fig. 3a, b), following the profile represented by the black solid lines in Fig. 4a, b. The ZDW is 1011 nm at the fiber input and reaches 1021 and 1030 nm at 14 and 29 m, respectively. Simulations and experiments are performed with an input pulse of 410 fs full width at half maximum (FWHM) duration centered around 1030 nm and a peak power of 46 W. In order to be consistent with experiments, we consider a slight chirp with a chirp parameter of C3:7. All parameters can be found in Billet et al. (2014). The simulation in Fig. 4a shows the emission of a first dispersive wave around 925 nm at a length of 5 m, in a similar fashion to what happens in uniform fibers (see Fig. 2). The soliton experiences a redshift due to the combined action of spectral recoil and Raman effect. The ZDW increases to 1021 nm at 1 m, which brings it closer to the soliton and therefore enhances the spectral overlap between the soliton and the normal GVD region. This initiates the emission of a second dispersive wave around 960 nm at this location in the fiber. The same process occurs a third time around 29 m, resulting in the emission of a third dispersive wave. Experiments reported in Fig. 4b are in excellent agreement with simulations and demonstrate the process of multiple dispersive wave emission from a single soliton.

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Cascading of Dispersive Waves The process of dispersive wave emission can also occur in a slightly different scenario in uniform fibers: a soliton experiencing a strong Raman-induced selffrequency shift can hit the second ZDW of a fiber (located at long wavelengths), which cancels the redshift and generates a dispersive wave across the ZDW, at longer wavelengths (Skryabin et al. 2003; Biancalana et al. 2004). Here we will study this process in an axially varying fiber (red curve in Fig. 3a, b) with two ZDWs separated by a region of anomalous dispersion in which a fundamental soliton is excited. The evolution of the second ZDW (the long wavelength one) is represented by the white solid lines in Fig. 5. It is constant over the first 7 m and then oscillates as a function of length. We consider input pulses of 340 fs FWHM duration centered around 1030 nm, with a chirp parameter of C1:5 and a peak power of 75 W. Figure 5a, b shows, respectively, the numerical simulations performed with Eq. 7 and experimental results. The input short pulse excites a fundamental soliton which initially experiences Raman-induced soliton self-frequency shift. Around 9 m, the ZDW has reached 1140 nm and is very close to the soliton so that a dispersive wave (labeled DW1 in Fig. 5a) is emitted across the ZDW, in the normal GVD region. At this point, the process is very similar to the one known in uniform fibers. The real novelty comes slightly after 10 m, when DW1 crosses the ZDW (which is increasing again

Fig. 5 (a) Numerical simulations and (b) experimental results in the spectral domain versus fiber length showing the emission of two cascaded dispersive waves in an axially varying fiber with two ZDWs. The evolution of the second ZDW (located at long wavelengths) with fiber length is represented by black solid lines

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at this point): as soon as DW1 crosses the ZDW, a new radiation (labeled CDW1) is generated at even longer wavelengths, around 1350 nm. A careful analysis of this process reveals that the continuously evolving GVD prevents the dispersive wave to strongly spread out in time (as expected in uniform fibers, see Fig. 2d) and allows to keep it relatively localized in time as a short pulse (Bendahmane et al. 2014). As a consequence, when crossing the ZDW, this pulse can emit another dispersive wave which we term cascaded dispersive wave (CDW1), in analogy with cascaded four-wave mixing processes. The exact same process occurs again farther in the fiber, around 15 m. The ZDW decreases again very close to the remaining of the soliton located slightly above 1100 nm so that a dispersive wave, DW2, is emitted. When DW2 crosses the ZDW slightly after, a cascaded dispersive wave, CDW2, is emitted. Experiments reported in Fig. 5b show again excellent agreement with numerical results and provide the evidence for the process of cascaded dispersive wave generation. We might also note that, similarly to the previous section, multiple dispersive waves (DW1 and DW2) have been observed from a single soliton, around the second ZDW.

Transformation of a Dispersive Wave into a Fundamental Soliton As mentioned above and deeply analyzed in Bendahmane et al. (2014), the cascaded dispersive wave process is due to the fact that the dispersive wave initially generated experiences a GVD that varies with length, which allows it to remain temporally localized as a pulse to initiate the generation of a cascaded dispersive wave. Here, we will study this process more into details. For that, we consider a soliton launched in the vicinity of the ZDW so that it emits a dispersive wave in the normal GVD region, similarly to the usual case illustrated in Fig. 2. Once the dispersive wave is emitted, the fiber parameters are changed so that the GVD at the dispersive wave wavelength becomes anomalous (green curve in Fig. 3a, b) and the evolution of the radiation is studied. The results are summarized in Fig. 6. The input pulse is transform-limited with a 140 fs FWHM duration. It is centered around 881 nm and has a peak power of 42 W. The spectral evolution versus fiber length (simulations in Fig. 6a and experiments in Fig. 6c) shows the emission of a dispersive wave around 840 nm but does not highlight much change after the GVD change occurring between 5 and 7 m. The simulated time domain simulation of Fig. 6b provides much more information about the dynamics. The soliton decelerates due to the combined effects of spectral recoil and Raman-induced self-frequency shift, and the dispersive wave, which starts spreading out, is located slightly behind it, as expected. Once it enters the tapered region (materialized by the two dashed lines), the dispersive wave accelerates due to changing dispersion and crosses the soliton. At the same time, it recompresses and further propagated as a short pulse in the remaining of the fiber, recalling the behavior of a soliton. In fact, further theoretical analysis reveals that the dispersive wave has indeed been transformed into a fundamental soliton (Braud et al. 2016) and propagates as such in the remaining anomalous dispersion region. Experimental autocorrelation measurements reported in Fig. 6d confirm this

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Fig. 6 (a, b) Numerical simulations and (c, d) experimental results showing the transformation of a dispersive wave into a fundamental soliton. (a) and (b) correspond, respectively, to the simulated spectral and temporal evolutions. (c) Measured spectral evolution versus fiber length. (d) Measured pulse duration around 840 nm (red full circles) and simulated one (black solid line). Black solid lines in (a) and (c) depict the ZDW. Dashed lines in (b) and (d) depict the limits of the tapered section

behavior: the pulse duration initially increases and then decreases after the tapered region, where they are perfectly fitted by square hyperbolic secant functions (Braud et al. 2016). These results show that a dispersive wave emitted from a soliton can be itself transformed into a fundamental soliton by carefully varying the longitudinal evolution of dispersion.

Emission of Polychromatic Dispersive Waves In the previous section, we have shown that a dispersive pulse initially traveling in normal GVD can become transformed into a fundamental soliton when entering a region with anomalous dispersion. Here we will study a reversed situation in which

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a fundamental soliton propagating in anomalous dispersion enters a region with normal dispersion. Fundamentally, the pulse cannot be a soliton anymore, and it is expected to linearly disperse. We will see that it can excite a so-called polychromatic dispersive wave (Milin et al. 2012; Kudlinski et al. 2015). Here, a fundamental soliton is excited by launching a transform-limited pulse with a 130 fs FWHM duration around 950 nm, with a peak power of 110 W. The fiber is uniform over the first 4.5 m, and then it is tapered down so that both ZDWs (depicted by solid lines in Fig. 7a, c) decrease until they join each other at 6 m. After this point, the fiber has all normal GVD all over the spectral range of interest here.

Fig. 7 (a,b) Numerical simulations and (c, d, e) experimental results showing the annihilation of a fundamental soliton into a polychromatic dispersive wave. (a) and (b) correspond, respectively, to the simulated spectral and temporal evolutions. (c) Measured spectral evolution versus fiber length. (d, e) Measured autocorrelation trace (open blue circles) and simulated ones (red solid line) at fiber length of (e) 4.5 m and (d) 6.5 m. Black solid lines in (a) and (c) depict the ZDWs. Dashed lines in (b) depict the limits of the tapered section

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The spectral dynamics, displayed in Fig. 7a, c for simulations and experiments, respectively, show the initial propagation of a soliton until 5.5 m, where it crosses the second ZDW and thus enters the normal GVD region. At this point, a spectacular spectral broadening occurs, indicating that the pulse experiences a nonlinear. In fact, as long as the soliton spectrum starts to cross the ZDW, the emission of a dispersive wave is initiated. Since the second ZDW keeps decreasing and crossing the soliton, there is a continuous emission of radiation into the dispersive wave. Because the GVD properties of the fiber continuously changes, the phase-matching relation (5) linking the soliton and the dispersive wave continuously changes too, resulting in the generation of a broad radiation termed polychromatic dispersive wave (Milin et al. 2012; Kudlinski et al. 2015). The process is as spectacular as in the time domain plot of Fig. 7b, where it can be seen that the significant broadening occurring at around 6 m is accompanied by a strong temporal broadening, which is consistent with the generation of a dispersive, strongly chirped pulse. This was confirmed experimentally by recording autocorrelation traces before and after the tapered section. At 4.5 m (Fig. 7e), the pulse has a duration of 113 fs (blue markers) and is well fitted by a square hyperbolic secant function, in good agreement with the simulation result (red solid line), which confirms its solitonic nature. But soon after the tapered section, at 6.5 m, the autocorrelation trace is much longer and has a much distorted profile, again in agreement with simulations. At this point, there is no clue of the presence of a soliton, which has therefore totally been annihilated into a polychromatic dispersive wave.

Generation of a Dispersive Wave Continuum In this section, we will investigate a complex scenario in which multiple solitons and dispersive waves are generated in an axially varying fiber. This results in the generation of a 500 nm supercontinuum exclusively composed of dispersive waves. For that, we use the same configuration as in section “Cascading of Dispersive Waves” except that we increase the peak power of the input pulse to 380 W. This exceeds the required power to form a fundamental soliton given by Eq. 3. In this case, the input pulse excites a higher-order soliton which immediately breaks up into several fundamental solitons (Beaud et al. 1987). Figure 8a, b shows, respectively, the simulated and experimental spectral dynamics recorded in this case. Three main solitons, labeled S1 to S3 , form here. The first one, S1 , is also the most powerful one (Lucek and Blow 1992) and therefore experiences the most efficient Raman-induced frequency shift. It rapidly hits the second ZDW (in black solid line) so that the frequency shift stops and a strong dispersive wave (labeled DW1 ) is emitted around 1600 nm. The soliton is frequency-locked around 1400 nm and hits the ZDW which starts to decrease at around 7 m. At this point, it emits a polychromatic dispersive wave (labeled PDW1 ) following the process described in section “Emission of Polychromatic Dispersive Waves.” The second soliton (S2 ) experiences a less significant Raman-induced self-frequency shift than S1 and hits

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Fig. 8 (a) Numerical simulations and (b) experimental results in the spectral domain versus fiber length showing the generation of a dispersive wave continuum in an axially varying fiber with two ZDWs. The evolution of the second ZDW (located at long wavelengths) with fiber length is represented by black solid lines

the ZDW around 8 m, where it is already decreasing. Therefore, it emits a broad polychromatic dispersive wave (labeled PDW2 ). Soliton S3 remains relatively far from the ZDW all over the propagation and thus does not contribute significantly to the emission of dispersive waves. Finally, the output spectrum between 1200 and 1600 nm is exclusively composed of dispersive waves and polychromatic dispersive waves generated from the two most powerful solitons.

Conclusion and Perspectives The perturbation of a fundamental soliton by third-order dispersion in optical fibers causes the emission of a resonant dispersive wave across the zero-dispersion point. In axially varying fibers, the guiding properties can be tailored as a function of propagation distance so that the ZDW continuously evolves as the soliton propagates. This allows to observe specific dispersive wave dynamics, such as the emission of cascaded, multiple, or polychromatic dispersive waves. Such axially varying fibers can also be used to control the properties of the soliton itself, such as harnessing its Raman-induced redshift (Judge et al. 2009; Bendahmane et al. 2013) or even inducing a blueshift (Stark et al. 2011).

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In periodic fibers, multiple quasi-phase-matched dispersive waves can also be observed thanks to the periodicity (Conforti et al. 2015, 2016), following a completely different mechanism than the one presented here (Conforti et al. 2015). More generally, this work illustrates remarkable robustness of fundamental solitons against various types of perturbations and the extremely rich nonlinear dynamics that these perturbations can induce.

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Nonlinear Waves in Multimode Fibers I. S. Chekhovskoy, O. S. Sidelnikov, A. A. Reduyk, A. M. Rubenchik, O. V. Shtyrina, M. P. Fedoruk, S. K. Turitsyn, E. A. Zlobina, S. I. Kablukov, S. A. Babin, K. Krupa, V. Couderc, A. Tonello, A. Barthélémy, G. Millot, and S. Wabnitz

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spatiotemporal Pulse Shaping in Multicore Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pulse Propagation in Multicore Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pulse Compression and Combining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlinear Pulses in Multimode Fibers for Spatial-Division Multiplexing . . . . . . . . . . . . . . . . Spatial-Division Multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlinear Propagation in Multimode Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Influence of Nonlinear Effects on the Propagation of Optical Signals . . . . . . . . . . . . . Raman Cleanup Effect and Raman Lasing in Multimode Graded-Index Fibers . . . . . . . . . . . Experimental Observations and Theoretical Models of Raman Cleanup Effect . . . . . . . . . Raman Cleanup Effect in Raman Fiber Amplifiers and Lasers . . . . . . . . . . . . . . . . . . . . . . GRIN Fiber Raman Lasers Directly Pumped by Multimode Laser Diodes . . . . . . . . . . . . . Combined Action of Raman Beam Cleanup and Mode-Selecting FBGs in GRIN Fiber Raman Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kerr Beam Self-Cleaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Models of Spatiotemporal Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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I. S. Chekhovskoy · O. V. Shtyrina · M. P. Fedoruk Novosibirsk State University, Novosibirsk, Russia Institute of Computational Technologies SB RAS, Novosibirsk, Russia O. S. Sidelnikov · A. A. Reduyk Novosibirsk State University, Novosibirsk, Russia A. M. Rubenchik Lawrence Livermore National Laboratory, Livermore, CA, USA e-mail: [email protected] S. K. Turitsyn Novosibirsk State University, Novosibirsk, Russia Aston Institute of Photonic Technologies, Aston University, Birmingham, UK e-mail: [email protected]

© Springer Nature Singapore Pte Ltd. 2019 G.-D. Peng (ed.), Handbook of Optical Fibers, https://doi.org/10.1007/978-981-10-7087-7_15

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Kerr Beam Cleanup in GRIN MMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kerr Beam Cleanup in Step-Index Active MMF with Loss or Gain . . . . . . . . . . . . . . . . . . Self-Cleaning in a MMF Laser Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

We overview recent advances in the field of nonlinear guided wave propagation in multimode fibers. It is only in recent years that the study of nonlinear optics in multimode fibers has experienced a revival of research interest. Nonlinear arrays of linearly coupled multicore fibers permit spatiotemporal reshaping and coherent combining of ultrashort optical pulses. Spatial-division multiplexing is an emerging technology for increasing the capacity of optical communication links, and the presence of nonlinear mode coupling requires a careful analysis. Multimode nonlinear fibers have a strong potential for the implementation of a new class of high-power fiber lasers. We describe experiments of Raman beam cleanup that permit to implement quasi-single-mode Raman fiber laser with multimode pumps. Next we discuss the recently discovered effect of Kerr beam self-cleaning, whereby a speckled signal at the output of a multimode fiber may evolve toward a bell-shaped transverse profile.

E. A. Zlobina · S. I. Kablukov Institute of Automation and Electrometry SB RAS, Novosibirsk, Russia e-mail: [email protected] S. A. Babin Institute of Automation and Electrometry SB RAS, Novosibirsk, Russia Novosibirsk State University, Novosibirsk, Russia e-mail: [email protected] K. Krupa Department of Information Engineering, University of Brescia, Brescia, Italy e-mail: [email protected] V. Couderc · A. Tonello · A. Barthélémy XLIM, UMR CNRS 7252, Université de Limoges, Limoges, France e-mail: [email protected]; [email protected]; [email protected] G. Millot ICB, UMR CNRS 6303, Université de Bourgogne, Dijon, France e-mail: [email protected] S. Wabnitz () Novosibirsk State University, Novosibirsk, Russia Department of Information Engineering, University of Brescia, Brescia, Italy National Institute of Optics INO-CNR, Brescia, Italy e-mail: [email protected]; [email protected]

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Introduction Multimode optical fibers (MMFs) have been investigated since the early 1980s, for their capacity of transmitting information over multiple spatial channels. However the presence of relatively large modal dispersion has largely prevented the use of MMFs for high-bit-rate transmissions over long-distance fiber-optic links. As a result, the long-distance fiber-optic infrastructure that has been built across the globe by operators and service providers over the past 30 years is virtually entirely based on single-mode optical fibers (SMFs). In recent years, there has been a revival of optical coherent communication systems, based on the technological development of fast, real-time digital signal processing techniques and hardware. This enables the digital compensation, in the electrical domain, of a large accumulated dispersion in the optical channel. Hence the technique of spatial-division multiplexing (SDM) using MMFs has received a renewed research and technological interest, as it may permit to overcome the capacity limitations of single-mode fibers for the next generation of fiber-optic networks. Guided modes of an ideal, longitudinally invariant optical fiber are orthogonal to each other, so that they can provide an alphabet for transmitting information. SDM may also be based on multicore optical fibers. Whenever the cores are sufficiently spaced apart, so that they do not interact via evanescent field overlap, a multicore fiber represents a simple way to spatially multiplex several independent information channels within the same optical fiber. On the other hand, nonlinear fiber optics has also been extensively studied both theoretically and experimentally since the 1980s. It is now a mature field of technology, and it has led to many technological breakthroughs such as ultrashort pulse fiber lasers, ultrafast optical signal processing devices, supercontinuum sources, and self-referenced optical frequency combs. However, basically all of these developments have involved single-mode optical fibers. Remarkably, it is not only until the last few years that nonlinear multimode fiber optics has started to receive the attention of researchers in a few leading groups. These initial studies have revealed a wealth of complex wave propagation phenomena, owing to the nonlinear coupling of spatial, temporal, and spectral degrees of freedom of the optical field. Besides their obvious interest from the viewpoint of basic science, nonlinear optical effects in multimode fibers hold the promise of several key technological advances, especially in the aforementioned fields of optical data communications, optical signal processing, high-power and brightness supercontinuum sources, and fiber lasers. In this chapter, we overview recent research progress among some of the most promising directions in the field of nonlinear multimode fiber optics. We consider first in section “Introduction” pulse propagation in arrays of linearly coupled cores in a multicore fiber with either circular or hexagonal symmetry. The resulting set of coupled nonlinear evolution equations provides a discrete-continuous analogous of the two- or three-dimensional nonlinear Schrödinger equation. Based on this analogy, it is shown that it is possible to control the spatiotemporal propagation of light pulses injected in the array of cores, in a way to combine their intensity into a single core at the fiber output.

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Next we investigate in section “Nonlinear Pulses in Multimode Fibers for Spatial-Division Multiplexing” nonlinear propagation of pulses in SDM transmission systems based on multimode fiber. We consider two essential cases of practical interest: weak- and strong-coupling regimes, in which the nonlinear propagation of optical signals is described by generalization of the so-called Manakov equations. The influence of nonlinear effects on the propagation of signals in a step-index multimode fiber is studied in the case of weak and strong coupling. These coupling regimes are compared with each other. We also investigate the impact of the number of propagating modes on the strength of nonlinear effects in multimode fibers. If sections “Introduction” and “Nonlinear Pulses in Multimode Fibers for Spatial-Division Multiplexing” are dedicated to describe the mode coupling and reshaping of short optical pulses, in sections “Raman Cleanup Effect and Raman Lasing in Multimode Graded-Index Fibers” and “Kerr Beam Self-Cleaning,” we describe essentially spatial nonlinear mode-coupling effects, activated either by stimulated Raman scattering or by the intensity-dependent refractive index of the fiber, known as Kerr effect. In both cases, we consider the highly multimode excitation of MMFs, which in linear conditions leads to a seemingly irregular spatial scrambling of the transverse intensity pattern at the fiber output owing to multimode interference. The presence of the Raman effect, on the other hand, permits under the condition of sufficiently high-power continuous wave (CW) excitation, to generate an essentially single-mode beam (typically close to the fundamental mode of the MMF) in the Stokes-shifted sidebands. As discussed in section “Raman Cleanup Effect and Raman Lasing in Multimode Graded-Index Fibers,” this effect is known as Raman beam cleanup, and it has found important applications to the development of high-power, single-mode Raman fiber lasers pumped by multimode diodes. We conclude the chapter with the overview in section “Kerr Beam Self-Cleaning” of recent experimental studies describing the discovery of Kerr beam self-cleaning in multimode optical fibers. In these experiments, using pulses with durations ranging from the nanosecond to the 100 femtosecond regime and propagating in the normal dispersion regime of a few meters long MMFs, it has been observed that, above a certain threshold power, the initially highly multimode beam evolves toward a bell-shaped transverse profile (sitting on a multimode, relatively low-power background), with a cross section very close to that of the fundamental mode of the fiber. The mechanism for this self-cleaning of light is the optical Kerr effect, as it is proved by the fact that the threshold power is inversely proportional to the fiber length. Thus beam self-cleaning is analogous to Raman beam cleanup, but it occurs also for the pump beam itself and not only for the Stokes-shifted wave. Initial demonstrations of this effect have involved graded-index, virtually lossless fibers. However it has also been reported that, in the presence of strong loss or gain, self-cleaning also occurs in a nearly step-index active ytterbium-doped MMF. Kerr self-cleaning has significant potential applications to high-power and ultrashort pulse fiber lasers, and initial experimental results of intracavity Kerr beam cleaning are presented.

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Spatiotemporal Pulse Shaping in Multicore Fibers Pulse shaping is commonly used in various fields of photonics and electronics; it is a process of changing the waveform of pulses for their better application. A lot of linear and nonlinear technics of pulse shaping in time and frequency domains were proposed in the last decades (Weiner 2000; da Silva et al. 1993; Boscolo et al. 2008; Andresen et al. 2011). Multicore fibers (MCFs) represent a set of waveguides located under common cladding. Along with multimode fibers, they offer the great possibility of spacedivision multiplexing (SDM) for high-capacity optical communications. They are now used for a great variety of applications in various areas of photonics (Minardi et al. 2010; Gasulla and Capmany 2012; Eilenberger et al. 2013; Tünnermann and Shirakawa 2015). Multicore fibers attract a lot of attention in the fields of optical communications (Richardson et al. 2013; Saitoh and Matsuo 2016; Igarashi et al. 2014; van Uden et al. 2014) and fiber lasers (Cheo et al. 2001; Ramirez et al. 2015; Hadzievski et al. 2015). MCFs, as an amplification media, are also attractive for linear beam combining applications in ultrafast laser systems. The increased effective area of MCF allows combining high-energy pulses with conserving a compactness of amplification scheme. In this approach, each core acts as an independent amplifying channel. In Lhermite et al. (2010), authors reported the co-phase combination of 49 beams at the output of a MCF. Experimental demonstration of pulse combining with the efficiency of 49% in the far field using a 7-core hexagonal MCF was presented in Ramirez et al. (2015). As mentioned in that paper, the deviation of residual phase differences, group delay differences among cores, and intensity variations decreased the efficiency, which theoretically can amount to approximately 76%. This discrepancy in amplification of each core is likely to result from MCF inhomogeneity. On the other hand, the cores in the MCF are coupled due to their proximity, reducing the phase mismatching among the cores. Recent studies have shown the possibility of using nonlinear effects in multicore fibers based on wave collapse for effective combination and compression, which offers new opportunities for nonlinear pulse shaping (Chekhovskoy et al. 2016). Initially, the idea to use the collapse for pulse compression in fiber arrays was proposed more than 20 years ago (Aceves et al. 1995); however, to build the fiber arrays is a technological challenge. MCF is an example of a fiber array with a specific distribution of a relatively low total number of cores, where the proposed ideas can be implemented. This type of nonlinear combining is substantially different from currently popular schemes of linear beam combining (Fan 2005) and can be advantageous for some other energy transfer and delivery applications. Here we discuss some latest results in nonlinear pulse combination and compression in multicore fiber and compare the effectiveness of ring and hexagonal 19-core fibers (Fig. 1a, d). The increase in the number of neighbors enhances the nonlinear effects and can make the compression and combination more robust and efficient. So the hexagonal structure should demonstrate improvements in performance as compared with ring core configurations. Massive numerical simulations were

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Fig. 1 Profiles of the 19-core MCF with hexagonal geometry (a) and the 19-core ring MCF (d). The dependence of the pulse compression factor (ratio of the initial width to the final one) (b, e) and the percentage of total energy combined in the central core at the first local maximum of the peak power of input pulse (c, f) on the parameters P and  in logarithmic scale of input Gaussian pulses for the hexagonal MCF and for the ring MCF

performed to determine the conditions of most efficient coherent combination and compression of the pulses injected into considered MCFs. We also discuss the influence of pulse phase perturbations and relative pulse delays on the combining scheme.

Pulse Propagation in Multicore Fibers The governing equations for a ring and square core configurations can be obtained in a similar way, so we consider the mathematical model on the example only of hexagonal multicore fibers. The electromagnetic field of optical pulses propagating along a N -core MCF with hexagonal core distribution can be well approximated by a superposition of modes E.x; y; z; t / D

X

An;m .z; t /Fn;m .x  xn;m ; y  yn;m /e i.ˇn;m z!t/ C cc;

(1)

n;m

where F gives the spatial mode structure and An;m is the complex envelope of the electromagnetic field in core number .n; m/, ˇn;m is the propagation constant, and the “cc” denotes “complex-conjugate.” In the limit of a weak-coupling approximation, the dynamic of the field envelope An;m of hexagonal MCF can be described

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using the discrete-continuous nonlinear Schrödinger equation (DCNLSE) (Mumtaz et al. 2013; Aceves et al. 2015; Chekhovskoy et al. 2016): i

@An;m ˇ2 @2 An;m  D @z 2 @t 2

X

Cn;m;k;l Ak;l   jAn;m j2 An;m ;

(2)

.k;l/¤.n;m/

where ˇ2 is the group velocity dispersion parameter and  is nonlinear Kerr parameter; both arep the same for all cores for simplicity. Using variables substitution An;m D exp.i 6z0 / C =Un;m with z0 D z=L, L D 1=C , where C is equal for all neighboring cores coupling coefficient (other couplings may be neglected) and t 0 D t =T , T 2 D ˇ2 =.2C /, the linear couplings between envelopes in neighboring cores can be represented as a discrete analog of second derivatives with respect to time (omitting the primes): i

@2 Un;m @Un;m C C .C U /n;m C jUn;m j2 Un;m D 0; @z @t 2

(3)

where .C U /n;m D Un1;m1 C UnC1;m1 C Un2;m C UnC2;m C Un1;mC1 C UnC1;mC1  6Un;m . This equation conserves the total energy ED

XZ n;m

1

jUn;m .z; t /j2 dt

(4)

1

and the Hamiltonian # 1 "ˇ ˇ 4 X Z ˇ @Un;m ˇ2 j jU n;m  ˇ ˇ H D dt: ˇ @t ˇ  .C U /n;m Un;m  2 n;m

(5)

1

For the ring core configurations, the governing equations take a form i

@2 Uk @Uk C C .UkC1  2Uk C Uk1 / C jUk j2 Uk D 0: @z @t 2

(6)

With a large number of cores and smooth pulse intensity distribution across the cores, the continuous limit of the discrete models (6) and (3) in the form of the well-known nonlinear Schrödinger equation (NLSE) can be used for a qualitative understanding of the system evolution. In particular, the pulse evolution in the ring MCF can be approximately described by continuous 2D NLSE for the function U .k; t; z/, considering the index k as a continuous variable: i

@U @2 U @2 U C 2 C C jU j2 U D 0: @z @t @k 2

(7)

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Equation (7) is equivalent to the NLSE that describes the self-focusing of light in various nonlinear media. The continuous analog can be used for insight into discrete system evolution. In the conventional theory of self-focusing governed by the NLSE, the initial wave distribution collapses into a singularity when the Hamiltonian H is negative or when the beam power exceeds the critical value Pcr . This value depends on the beam shape and is minimal for the Townes mode. In our case, R the role of power is played by the total energy injected into the MCF E D dt d njUn j2 . When the input energy exceeds the critical value E0 , (for thep Townes beam, E0 D Ecr D 4, in terms of the dimensional variables Ecr D 4 C ˇ2 =.2 2 /), making H < 0, the intensity distribution is self-compressed over k and t . We can expect that the injected MCF pulses distributed over the cores with smooth maxima will be focused into a few cores around the maxima with simultaneous pulse compression. When the energy is concentrated into a few cores, the discreteness of the cores arrests further compression. When the input energy E  4, the distribution breaks into a few collapsing clusters with E  4 (similar to filamentation in the continuum limit). In every cluster, the compression and combining take place, but the location of the peaks is difficult to predict, and this situation is not practical for the goals of beam combining or pulse compression. The continuous version of Eq. (3) takes the form of 3D NLSE:

i

@2 U @U @2 U @2 U C 2 C C C jU j2 U D 0: @z @t @k 2 @l 2

(8)

An increase in the number of neighbors enhances the nonlinear interaction and makes collapse possible even for positive values of H (Kuznetsov et al. 1995). In this case, we can expect that the injected MCF pulses will be focused into a few central cores with simultaneous pulse compression. Nonlinear systems described by Eq. (8) have stronger collapsing features, and an MCF with 2D configuration of cores could demonstrate better compression results, which will be shown later. In the 3D situation, the collapse is “weak” (see definitions in Kuznetsov et al. 1995), and the energy involved in the compression processes decreases (Zakharov and Kuznetsov 2012). In this case, the optimal compression and combining are reached in a transient regime, and the choice of parameters is not universal. In the next subsection, we will present the results of the modeling of compression and combining in both situations and will examine the selection of the optimal parameters.

Pulse Compression and Combining For 19-core hexagonal MCF, we simulated the dynamic of Gaussian pulses Un;m .t / D

p P exp .t 2 = 2 /

(9)

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injected in all cores of this fiber to define values of amplitudes P and widths  providing the most efficient pulse combining and pulse compression. The dependencies of the basic compression parameters are obtained for the range of parameters P 2 Œ0:05I 1000,  2 Œ0:05I 230. We tracked the first local maximum of the peak power of the pulses propagating along the MCF. This event corresponds to the pulse compression for some domain of parameters P; . The pulse dynamic was modeled by the system (3). For numerical simulations we propose to use the generalization of the split-step Fourier method with the Padé approximation or finite-difference compact schemes (see Chekhovskoy et al. 2017). We compared the domains with efficient combining and compression for a 19core hexagonal MCF and for a 19-core ring MCF (Fig. 1). Recently it has been shown that initial modulation of initial Gaussian pulses improves the combining scheme stability for the ring MCF if the phases of injected pulses are perturbed. Therefore now we use pulses in the form Uk .t / D

p

P exp .t 2 = 2 /.1 C 0:3 cos.2k=N /;

(10)

where N is the number of cores, k D 1; : : : ; N , with a smooth distribution of the peak power with a maximum in N -th core. The maps of combining and compression performance for the MCFs under consideration are presented in Fig. 1. Figure 1b, e shows the dependence of the total energy percentage of all pulses combined at the first maximum of the peak power of the pulse in central core (combining efficiency). Due to the symmetry of the problem, both combining and compression take place in the N -th core for the ring MCF and in the (0,0)-core for the hexagonal fiber. We see that the optimal conditions for combining and compression in the case of hexagonal fiber are very different from the results for the ring core distribution. Efficient combining takes place in a much broader range of parameters in the vertical band, insensitive to the pulse duration. The efficiency of combining is comparable with the ring MCF but much less sensitive to variation of the initial parameters. Using ring 19-core fiber, it is possible to get combined pulse having about 80% of total injected in fiber energy at the distance along fiber z D 65:9. The maximum combining efficiency for a 19-core hexagonal MCF equals 80.9% (Fig. 2); the combined pulse can be obtained at the distance z D 2:07. In contrast to the ring core configurations, a wide region in the plane of parameters .P;  / exists, where a part of the energy in the central core at the compression moment exceeds 70% of the initial energy E. The presence of this region allows us to obtain well-compressed pulses having most of the total energy E, which is of great practical importance. However, in this case a substantial part of the energy goes into the wings rather than into the central peak of the compressed pulse. Figure 1c, f presents the dependence of the compression factor of a pulse (ratio of the initial full width at half maximum (FWHM) to the final one). The blue area corresponds to the pairs of the parameters P and  , for which there is no pulse compression or the initial pulses (10) break into clusters as a result of the modulation instability. It is interesting that isolines of pulse compression factor in Fig. 1c, f

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Fig. 2 The intensity dynamic of the input Gaussian pulses (9) with the parameters P D 0:36 and  D 1:69 injected in all cores of a hexagonal 19-core MCF (best combining) (a). The input intensity distribution in the central core (dashed red line) and the distribution at compression point (solid blue line) in logarithmic scale (b). 80.9% of total input energy E is combined in the core (0,0). The pulse width (FWHM) is reduced by 7.3. The peak power increases in 103.4

Fig. 3 The intensity dynamic of the input Gaussian pulses (10) with the parameters P D 0:0545 and  D 184:0 injected in all cores of the ring 19-core MCF (best compression) (a). The input intensity distribution in the 19th core (dashed red line) and the distribution at compression point (solid blue line) are shown in logarithmic scale (b). 10.9% of total input energy E is combined in the 19th core. The pulse width (FWHM) is reduced by 720.4. The peak power increases in 314.8

correlate with isolines of the ratio of the dispersion length LD D  2 =jˇ2 j and the nonlinear length LNL D 1=. P /. The zone of optimal compression for the 19-core ring MCF is the stripe narrowing toward high total energies and is confined by the level LD =LNL  3000. If LD =LNL > 3000, a large nonlinearity destroys a smooth pulse shape before the compression point. It is worth noting that the effective compression and effective combining cannot be reachable simultaneously. The maximal compression was obtained by the initial pulses with LD =LNL  3000 and approximately equals 720, the compressed pulse has a smooth profile (see Fig. 3). The distance to the compression point is equal to 141.08 in this case. The optimal parameters for the compression using hexagonal MCF differ from the results for the ring fiber. Using the 19-core hexagonal fiber, a maximal pulse compression up to 256 times can be achieved. In contrast to the combining pattern,

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at the point of maximal compression, a significant fraction of energy is left in the neighbor cores. The peak power increases greatly at the compression point. The best compression occurs in the case of a high-power input pulse. Too much nonlinearity (LD =LNL  4000) destroys the pulse shape before the compression point. On the other hand, it is difficult to define the optimal compression case in the presence of high nonlinearity. The pulse compression factor close to its maximal value can be obtained for different pairs of parameters P; , and the distance to these compression points decreases with growth of the parameter P . The compression distance is sufficiently small for the 19-core hexagonal MCF (z  0:5). Thus, the stronger nonlinear interaction for a hexagonal MCF does not increase the efficiency of compression (combining) but greatly reduces the required distance and increases the process robustness. Therefore, the hexagonal MCFs have an advantage of being used as a base of the optical pulse compressor device. It is critically important to control the relative phases of all injected pulses when using the linear pulse combining. In the case of nonlinear combining scheme, the phase-matching requirements can be relaxed. The results of a stability analysis for the first considered regime giving the best combining for the 19-core hexagonal MCF are presented in Fig. 4. We modeled the phase perturbations as uniformly distributed on the segment Œıp I ıp  with a random function Cıp , so the initial pulses were UQ n;m .t / D Un;m .t / exp ŒiCıp :

(11)

The parameter ıp varied from 0 to . All computed values were averaged on 1000 launches for every value of ıp . The calculations showed that the combining scheme is stable for ıp 2 Œ0I =5. If ıp > =5, the pulse compression can occur in cores other than the central core (Fig. 4a). The distance to the compression point and the standard deviations from the mean value are shown in Fig. 4b. The limit values of temporal pulse delays that do not affect the combining and compression process were also determined. The random pulse delays were modeled as uniformly distributed on the segment Œıt I ıt  with the random function Cıt , making perturbed initial pulses UQ n;m .t / D Un;m .t  Cıt /:

(12)

The simulations showed the stability of the combining scheme for ıt 2 Œ0I , where  D 1:8 is the width of the injected pulses (see Fig. 4c, d). Summing up, an effective nonlinear scheme for combining and compressing pulses using a 19-core ring and hexagonal MCF can be realized. In comparison with a ring geometry, the enhanced nonlinear interaction in the hexagonal fiber does not result in more efficient compression but makes the system shorter and, therefore, more practical. The limits of stability of this scheme to random phase perturbations of the input pulses were found.

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Fig. 4 The influence of phases mismatches and pulse delays on the compression scheme. The dependence of the probability of compression among cores of the 19-core hexagonal MCF on the phase-mismatch parameter ıp (a) and on the pulse time delay parameter ıt (b) for initial Gaussian pulses with amplitude P D 0:53 and width  D 1:8. The corresponding mean value and standard deviation of the distance to the peak power maximum (b, d) averaged on 1000 launches

Nonlinear Pulses in Multimode Fibers for Spatial-Division Multiplexing Spatial-Division Multiplexing Fiber-optic communication lines are by far the most effective information systems for transmitting large amounts of data over long distances at high speed. In modern systems based on a standard single-mode fiber (SSMF), all the available degrees of freedom (time, frequency, phase, and polarization) can be used to modulate and multiplex signals. Different technologies were called upon to achieve a data transfer rate of over 100 Tb/s for SSMF systems (Qian et al. 2011), but further increase in the capacity of such systems is difficult due to various limitations (Essiambre et al. 2010). At present, there is an imbalance between the growth of global information traffic, which is about 40% per year, and the growth of the total transmission

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capacity of modern fiber-optic links, which is about 20% a year. And if no new technology providing a significant increase in the data transfer rate is developed in the near future, we may face the problem of the traffic volume exceeding the capabilities of modern data transmission systems. The spatial-division multiplexing (SDM) is currently considered as a promising technological pathway for further increasing the transmission capacity of optical networks (Winzer and Foschini 2011). Such systems require the development of fibers that support propagation in several spatial modes or cores. These fibers include multimode fibers (MMF), multicore fibers (MCF), and hollow-core fiber (HCF) (Essiambre et al. 2013). Due to a number of advantages (cost compared to MCF and HCF, ease of installation, integrability in most of existing data transmission systems), multimode communication systems are by far the most promising for increasing the transmission capacity of optical networks. Multimode fibers are fibers with one core of a sufficiently large diameter to support more than one spatial mode. The number of spatial modes supported by MMF grows rapidly depending on the core diameter and may number in the hundreds. The use of multimode fibers allows significantly increasing the capacity of optical networks by simultaneously transmitting signals over different fiber modes. However, new effects influencing the transmitted signals appear at a simultaneous use of several modes, namely, linear mode coupling, differential mode delay (DMD) (Ho and Kahn 2011), and inter-mode nonlinear effects (Rademacher et al. 2012). Furthermore, the nonlinear effects are one of the major factors that limit the capacity of data transmission systems (Ellis et al. 2010). Therefore, for a successful use of multimode fibers as a method of increasing the data transfer rate, it is necessary to investigate and understand each of these effects.

Nonlinear Propagation in Multimode Fibers The total electric field in a multimode fiber can be written as a sum over M of different spatial modes in the frequency domain (Mumtaz et al. 2013):

Q E.x; y; z; !/ D

M X

p e iˇm .!/z AQm .z; !/Fm .x; y/= Nm ;

(13)

m

where AQm .z; !/ D ŒAQmx .z; !/; AQmy .z; !/T is the Fourier transform of the slowly varying envelope of the field in the time domain of the mth mode. This expression includes the slowly varying amplitudes of both polarization components of the spatial mode m with the spatial distribution Fm .x; y/ and the propagation constant ˇm .!/. The normalization constant Nm represents the power carried by the mth and can be expressed as Nm D 12 "0 nN eff cIm , where Im D R R mode 2 nN m =nN eff Fm .x; y/dxdy, "0 is the vacuum permittivity, nN eff is the effective index of the fundamental mode, and nN m is the effective index of the mode m.

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Here we consider two essential cases of propagation in multimode fibers, which are of practical interest: weak- and strong-coupling regimes. In the weak-coupling regime, the coupling between different spatial modes is weak compared to the coupling between two polarization components of one spatial mode. In addition, in this case the fiber length is not much longer than the correlation length (Kahn et al. 2012). In the strong-coupling regime, both types of coupling are of the same order, and the fiber length far exceeds the correlation length. In practice, some spatial modes of the multimode fiber can be weakly coupled, while others can be coupled more strongly. The equations of nonlinear propagation of optical signals in multimode fibers are obtained from the Maxwell’s equations using the approach of Kolesik and Moloney (2004). For more details of deriving the equations of propagation, see Mumtaz et al. (2013), Poletti and Horak (2008), and Mecozzi et al. (2012). In this case, the nonlinear propagation of signals in multimode fibers in the weakcoupling regime is described by the following Manakov equation (Mumtaz et al. 2013): 0 1 X @Ap @Ap ˇ2p @2 Ap 8 4 D i  @fpppp jAp j2 C fmmpp jAm j2 A Ap ; C ˇ1p Ci @z @t 2 @t 2 9 3 m¤p

(14) where Ap .z; t / is the slowly varying envelope of the pth mode in the time domain; ˇ1p and ˇ2p are the inverse group velocity and the group velocity dispersion of the pth spatial mode, respectively; and  D !0 n2 =cAeff is the nonlinear parameter of fiber, where n2 is the nonlinear refractive index for glass, Aeff is the effective area of the fundamental mode at the central frequency !0 , and flmnp are the coefficients of nonlinear coupling between spatial modes, which have the following form: flmnp D

Aeff .Il Im In Ip /1=2

Z Z Fl Fm Fn Fp dxdy:

(15)

In the case of strong-coupling regime, the equation of signal propagation is described by the following Manakov equation (Mecozzi et al. 2012): @AN ˇ 00 @2 AN @AN N N 2 A; C ˇ0 Ci D i  jAj @z @t 2 @t 2

(16)

where D

M X 32 fkkl l ; ı kl 2 6M .2M C 1/

(17)

kl

AN is the vector of slowly varying mode amplitudes, ˇ 0 is the average inverse group velocity, and ˇ 00 is the average group velocity dispersion.

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It should be noted that in the weak-coupling regime, the signals in different spatial modes propagate with different velocities, that is, the coefficients ˇ1p will be different for different modes. In the strong-coupling regime, the parameter ˇ 0 is a scalar, so the modes will propagate at the same velocity. In addition, in the case of weak coupling, the nonlinear interaction will increase with an increase in the number of modes, since the sum on the right-hand side of Eq. (14) grows. In the case of strong mode coupling, the nonlinear parameter  (17) in Eq. (16) will decrease, since the denominator has a term of the order of M 2 .

The Influence of Nonlinear Effects on the Propagation of Optical Signals The simulated transmission link is shown in Fig. 5 and consists of a transmitter, 10 spans of MMF, and a receiver. Each transmitter generated 16-QAM modulated raised cosine pulses at a symbol rate of 32 Gbaud, with a roll-off factor of 0.2 and an oversampling factor of 16. The central wavelength of the emitted signal band was at  D 1550 nm. The generated signals were subsequently launched into a transmission link that consisted of 10 spans of 100 km step-index multimode fiber each. The MMF parameters were refractive index of the core nco D 1:454, refractive index of the cladding ncl D 1:444, core radius a D 7 m, and fiber loss ˛ D 0:2 dB. Such fiber supports the propagation of four modes without taking into account the degenerate ones (LP01, LP11, LP02, LP21). Differential modal delays (DMD), dispersion parameters (D), and effective mode area for four modes are shown in Table 1.

Fig. 5 Scheme of simulated transmission link

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Table 1 DMD, D, and Aeff of four guided modes

Table 2 Coefficients of nonlinear coupling between spatial modes

LP01 LP11 LP02 LP21

– LP01 LP11 LP02 LP21

DMD [ps/km] 0 4:2 6:1 7:9

LP01 1

D [ps/(km–nm)] 22:5 23:2 3:3 17:2

LP11 0:6294 0:9932

LP02 0:6871 0:3283 0:8474

Aeff [m2 ] 102 103 121:4 122

LP21 0:4068 0:5585 0:2934 0:8425

The coefficients of nonlinear coupling between all spatial modes with a step-index profile of the refractive index are shown in Table 2, the lower part of which is not filled on the grounds of fmmpp D fppmm . An EDFA of 4.5 dB noise figure was used to fully compensate the signal attenuation. The equations of propagation were solved numerically with a typical symmetrized split-step Fourier method. After transmission through the channel, the signals were coherently detected. Then a linear equalization was used for ideal compensation of chromatic dispersion effects. After down-conversion to one sample per symbol, the nonlinear equalization was performed based on least mean square (LMS) algorithm. To analyze the system performance, we estimated the Q-factor, associated with BER by Q D 20 log10

hp i 2 erfc1 .2BER/ ;

(18)

where BER represents the average bit error rate for all modes, obtained by direct error counting. In the study of the influence of nonlinear effects, first we compared the modecoupling regimes. Figure 6 shows the dependence of the quality parameter Q-factor on the initial signal power for the strong- and weak-coupling regimes for data transmission using four spatial modes. As can be seen from the figure, the weakcoupling regime shows better performance in comparison with the strong-coupling regime. This can be explained by the fact that at a strong mode coupling, the signals propagate with the same velocities, and in the course of propagation, the nonlinear interaction between them remains high (Fig. 7 left). In the case of weak coupling, the signals in different modes propagate with different velocities. This leads to the fact that the peaks of the pulses along the propagation move relative to each other (Fig. 7 right), and the nonlinear interaction between them decreases. If we consider the weak-coupling regime in which the signals in different modes propagate with the same velocities (blue line in the Fig. 6), then the performance will be worse than in the strong-coupling regime. This is due to the fact that at a strong

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Fig. 6 Comparison of mode-coupling regimes

Fig. 7 Propagation of the signals in different spatial modes with the same (left) and different (right) velocities

mode coupling, the nonlinear parameter  (17) will be smaller than the nonlinear part in the weak-coupling regime (14). To investigate the effect of a differential mode delay on the transmission performance, the dependence of the bit error rate on DMD has been found. Figure 8 shows the dependence of BER on DMD for signal propagation in the case of weak coupling using two modes. As can be seen from the figure, the number of transmitted errors decreases with the increase of differential mode delay. However, BER decreases to a certain limit, corresponding to the case when in the course of propagation the optical pulse in one spatial mode reaches the neighboring pulse in another mode. We also investigate the influence of nonlinear effects with an increase in the number of modes in the strong- and weak-coupling regimes. Figure 9a shows the dependence of Q-factor, averaged over all used modes, on the initial signal power for the propagation in one, two, three, or four modes at strong coupling. In this case, an increase in the number of modes leads to a worse transmission performance, but

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Fig. 8 BER as a function of DMD for signal propagation in the case of weak coupling using two modes

the decrease in the Q-factor slows down for a large number of modes. This is due to the fact that in the strong-coupling regime, the nonlinear parameter  (17) will decrease as the number of used modes increases. Figure 9b shows the dependence of the Q-factor on the initial signal power for optical signal propagation in one, two, three, or four modes in the weak-coupling regime. As can be seen from the figure, the cases of one and two modes show approximately the same Q-factor; however, when adding the signal, propagating in the third mode LP02, the transmission performance declines significantly. This is due to the fact that for LP02, the value of the dispersion parameter is small (3.34 ps/(km–nm)), and the signal propagating in this mode undergoes a smaller dispersion broadening and, therefore, is more susceptible to nonlinear distortions. Thus, the input of the LP02 mode to the Q-factor averaged over all used modes decreases the system performance. A slight improvement may be reached by adding the fourth mode LP21 with rather large dispersion parameter. In this case, the value of the Q-factor is greater than in the case of three modes.

Raman Cleanup Effect and Raman Lasing in Multimode Graded-Index Fibers The Raman beam cleanup effect is known as the effect of stimulated Raman scattering (SRS) of a low-quality pump beam, which produces a redshifted beam with much better beam quality. Laser generation is also possible in passive fibers owing to the SRS-induced amplification of the scattered light. A value of the socalled Stokes shift is defined by the vibration quanta amounting to 13 THz in standard telecommunication silica fibers (Stolen et al. 1972). Similar effect was also

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Fig. 9 Q-factor as a function of initial power for signal propagation in the case of strong (a) and weak (b) coupling using one, two, three, and four modes

observed in another dissipative process such as stimulated Brillouin scattering (SBS) (Rodgers et al. 1999; Russell et al. 2001; Lombard et al. 2006). There are several important distinctions between the SBS and SRS. The first and most important difference is the gain bandwidth (100 MHz for SBS and 3–10 THz for SRS). This makes SBS usable only for narrowband lasers, whereas SRS is important for broadband lasers. Second, SRS amplifies both the backward and forward scattered light, whereas SBS supports backward scattering only. Third, the Stokes shift for SRS is by three orders of magnitude larger than that for SBS. Another consequence of the large Stokes shift is that, the SRS-induced Stokes beam intensity profile will

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deviate from that for the pump beam more quickly. Thus the fiber length required for the beam cleanup is considerably shorter than that required for SBS cleanup.

Experimental Observations and Theoretical Models of Raman Cleanup Effect In the first experimental demonstrations, the field pattern of the Stokes waves generated in the SRS process had a ring shape of much smaller diameter than the pump one, as it was observed independently in bulk media (Komine et al. 1986; Goldhar et al. 1984; Hanna et al. 1985; Reintjes et al. 1986) and multimode gradedindex (GRIN) fibers (Baldeck et al. 1987; Grudinin et al. 1988; Nesterova et al. 1981) in the 1980 s. Van den Heuvel has developed a numerical model of SRS beam cleanup in bulk media which explains the process in terms of the balance between amplification and diffraction (van den Heuvel 1995). In the case of multimode GRIN fibers, the researchers used picosecond pump radiation at wavelengths of 532 or 1064 nm with kW-level peak power. As a consequence, most of them explained the observed phenomenon as a result of self-focusing. Chiang (1992) supposed that the significant improvement of spatial beam quality of the Stokes component generated via SRS can be explained in terms of mode competition between the various Stokes transverse modes of a graded-index fiber. He questioned the self-focusing mechanism because the nonlinear part of refractive index calculated by him was too small to modify the field confinement in the fiber (Chiang 1992, 1993). In order to check the assumption of mode competition, Chiang launched nanosecond pulses from a dye laser operating at 585 nm into a 30-m multimode graded-index (GRIN) fiber and observed that, under optimum pump launching conditions, the generated four Stokes waves propagate in the fundamental LP01 mode. When the angle and the position of the input fiber end relative to the pump beam were slightly changed, several low-order modes and their combinations were preferentially excited at the Stokes wavelengths (Fig. 10a). For a 1-km fiber, higher-order Stokes waves were generated in the fundamental mode even if the first Stokes wave contains a mixture of a large number of modes (Fig. 10b, c). So, the process of cascaded SRS in the fiber is accompanied by stripping off the high-order modes. Chiang explained this effect by increasing mode coupling in the long fiber which results in a steady-state mode distribution favoring low-order modes. This effect was also noticeable in Pourbeyram et al. (2013), where nanosecond radiation at 523 nm generated multiple cascaded Raman peaks extending up to 1300 nm at propagation in 1 km 50/125 multimode GRIN fiber. Although the non-diffraction-limited pump beam excites multiple modes, all Stokes peaks comprise only low-order modes. For example, the sixth Stokes component at 610 nm consists of the fundamental mode only. However, the higher-order Stokes waves start to propagate in the lower transverse modes most likely due to the combined effect of SRS and non-degenerate collinear four-wave mixing (FWM). Chiang also proposed a first simple model of SRS beam cleanup in multimode fibers based on overlap integral of the pump field modes and the Stokes field

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Fig. 10 (a) Images of the first Stokes mode patterns obtained in a 30-m fiber with the pump filtered out; images of the dispersed output from a 1-km fiber showing two situations: (b) the LP01 mode evolves at all the Stokes shifts, (c) the LP01 mode evolves in the second Stokes shift, while the first Stokes shift contains a number of modes

modes (Chiang 1992). He considered a short multimode GRIN fiber where the mode coupling was negligible. The main drawback of the model is that the intensity profiles of step-index fiber modes were used instead of graded-index fiber modes, in spite of that the SRS beam cleanup is absent for step-index fibers. Nevertheless, the normalized overlap integral in the calculation has a maximum value when the excited mode and the evolving mode were the same. Therefore, Chiang concluded that the preferentially excited mode usually has the highest Raman gain. This model explains the results obtained in the experiment with the 30-m MM GRIN fiber (Fig. 10). Following Chiang, Polley et al. (Polley and Ralph 2007) calculated the effective Raman gain for the lowest-order pump and signal mode pair, considering elements of the Raman gain matrix Gij for the j -th signal mode gain due to the i -th pump mode with power Pi over the effective length Leff :  Gij D exp

gR Aeff



! Pi Leff :

(19)

ij

The effective Raman gain coefficient, i.e., the ratio of Raman gain coefficient gR to the effective area, was found for each pump and signal mode pairs .i; j / as

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gR Aeff

 ij

RR DRR

gR .r/

2 i .r; /

2 i .r; /drd

RR

2 j .r; /drd 2 j .r; /drd

;

(20)

where i and j are the transverse electric field profiles for the j -th signal and the i -th pump mode, respectively. The field of confined LPmp modes in a weakly guiding graded-index fiber is given by: mp .m/

   .m/  D cos .m / Rm Lp1 VR2 exp VR2 =2 ;

(21)

where Lp1 represents the associated Laguerre polynomial; R D r=a, where a is the radius of the fiber core, V is the normalized frequency of the fiber. Therefore, the overlap integral differs from that calculated by Chiang. Nevertheless, it has the highest value for the pair of LP01 pump and signal modes. The effective Raman gain computed for the lowest-order mode (LP01 ) in 62.5 and 50 m MM GRIN fibers appeared to be comparable to the effective Raman gain for standard SMF, amounting to 0.5 W1 km1 . In a graded-index fiber, the doping density of GeO2 varies with radial coordinate and can be estimated from the refractive index profile. Therefore, the Raman gain coefficient, which increases with the doping, is also a radial function. Compared to SMF, the higher Raman gain coefficient due to the higher GeO2 doping completely compensates for the larger effective area of the lowest-order MMF modes. Finally, Terry et al. explained why beam cleanup does not occur in step-index fibers (Terry et al. 2007a). Their model of SRS beam cleanup describes beam cleanup in graded-index and step-index fibers in terms of their overlap integrals for a range of launching conditions. The initial power in the Stokes beam was considered as uniformly distributed over the transverse modes at single longitudinal mode, like the pump one. It was also assumed that each Stokes mode interacts with one pump mode only. The values of the calculated overlap integrals showed that the overlap of the LP01 pump mode with the LP01 Stokes mode is greater than the overlap of any other pair of intensity patterns. Moreover, the largest values of the overlap integral are reached when a pump mode overlaps with its corresponding Stokes mode. It becomes progressively smaller for higher-order transverse modes. According to this simple description, mode competition in a graded-index fiber favors the LP01 Stokes mode that corresponds to Polley’s table (Polley and Ralph 2007). In a stepindex fiber, the overlap of the higher-order transverse modes with their respective pump modes was greater than the overlap of the LP01 Stokes mode with the LP01 pump mode, which corresponds to Chiang’s table (Chiang 1992). Therefore, mode competition in a step-index fiber does not favor LP01 mode. Alternative theory of SRS beam cleanup was proposed by Murray et al. (1999). It is based on the central limit theorem in statistics and the theory of linear systems. The theorem states that repeated convolutions (or correlations) of arbitrary functions asymptotically produce a Gaussian function. By decomposing the input pump and Stokes beams in terms of their plane-wave components, Murray et al. expressed the driving nonlinear polarizations for SRS as a series of autocorrelation and

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cross-correlation operations between the plane-wave components of the beams. The resulting polarization is a function having a prominent central peak in the transverse plane, which favors amplification of the Stokes beam in the fundamental GRIN fiber mode. The theory is also applicable to the case of beam cleanup via stimulated Brillouin scattering in fibers. Summarizing the above, the beam cleanup property of SRS is commonly attributed to the preferential amplification of the fundamental mode of the Stokes beam relative to the higher-order modes through its better overlap with the multimode pump beam in GRIN fibers. Since all modes in the Stokes waves grow exponentially with their own gains, the difference in power between any two Stokes modes with different gains also grows exponentially with distance. Now consider several practical applications of the SRS cleanup effect. In 2002, Russell et al. performed a SRS beam cleanup experiment using a 300m-long 50 m graded-index fiber (Russell et al. 2002). In this experiment, the multimode fiber was pumped with a frequency-doubled Q-switched Nd:YAG laser at 532 nm. While the transmitted pump beam has poor beam quality (M2 D 20:7), the Stokes beam has a greatly improved beam quality (M2 D 2:4). The Stokes beam contains multiple Stokes shifts spanning over 100 nm. In order to calculate M2 , Russell et al. assumed that all Stokes waves have the first Stokes wavelength. Studying the dependence of the Stokes beam spot size on the GRIN fiber length, they found a transition point between long and short fibers, in which the beam cleanup properties were lost and higher-order modes were excited. Fibers longer than 100 m generate Stokes beams with Gaussian intensity profile (Fig. 11a). Under 100 m, the Stokes beam was found to fluctuate between a Gaussian-like profile and a multi-lobe profile (Fig. 11b), in agreement with Chiang’s observations (Chiang 1992). Russell et al. (2002) also proposed to apply beam cleanup properties of SRS in multimode fibers for brightness enhancement and scaling of common laser beams through the beam cleanup and combining, respectively. First successful combination and cleanup of four laser beams via stimulated Raman scattering in GRIN fibers was demonstrated by Flusche et al. (2006). Four coherent pulsed laser beams split off from a Q-switched Nd:YAG laser were combined by multi-port fiber combiner and then launched into a 2.5-km-long GRIN fiber with 100 or 200 m core diameter. Due to the long fiber and high peak power, cascaded Stokes shifts up to 1.5 m were generated through combined effects of SRS and non-degenerate collinear FWM. Despite of different beam quality of the transmitted pump beam in 100 m .M2 D 26/ and 200 m .M2 D 42/ GRIN fibers, the Stokes beam has a good quality

Fig. 11 Pump (left) and Stokes (right) intensity profiles for 300 m (a) and 75 m (b) fiber

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with nearly the same M2 value of 2.7. The SRS beam cleanup was also studied depending on the fiber length: M2 of the Stokes beam slightly increases from 1.5 to 2 with shortening the fiber from 1.3 to 0.4 km. In addition to the beam combination, the SRS beam cleanup effect was also applied to Raman fiber lasers (Baek and Roh 2004; Terry et al. 2007b), amplifiers (Polley and Ralph 2007; Rice 2002; Terry et al. 2008), and even to multimode fiber communication links (Polley and Ralph 2007). In the last case, it improves intersymbol interference for 10-Gb/s data transmission in multi-kilometer multimode fiber link.

Raman Cleanup Effect in Raman Fiber Amplifiers and Lasers In Raman fiber amplifiers (RFAs), as predicted by Rice (2002), the lowest-order Stokes mode launched into the core of graded-index multimode fiber grows at the expense of higher-order Stokes modes. The RFA could therefore use a multimode pump beam to amplify a single-mode Stokes signal. Because the multimode pump beam can be efficiently coupled into a multimode fiber, an RFA based on a multimode graded-index fiber offers an intriguing route to creating a highly efficient RFA with single-mode output. Terry et al. (2008) refuted this assumption. They modeled the beam quality of the Stokes output by considering the relative gain of the Stokes modes of the fiber. The model was based on a set of coupled differential equations that describe the interaction of the pump and Stokes components. It is evident that in the absence of an external Stokes input, spontaneous Raman scattering in a long fiber uniformly seeds all the transverse Stokes modes of the fiber. Since all Stokes modes are seeded with equal amounts of power, the output power of a given Stokes mode depends upon its gain. In the case of RFA, the output power of a given Stokes mode depends on both its gain and its input power. So, it is possible that the Stokes modes with low gain and high initial power dominate over modes with high gain and low power. The most promising application of SRS beam cleanup relates to the power scaling in high-power Raman fiber lasers (RFLs). Relatively low Raman gain, as compared to the gain in active fibers, requires a much larger length of passive fiber exceeding hundreds of meters in conventional RFLs. The power of single-mode LDs is limited by 1 W level that is comparable with the RFL threshold. That is why RFLs are usually pumped by high-power Yb- or Er-doped fiber lasers with single-mode output coupled directly to the core of a passive Raman fiber. To ensure the single-mode character of the output beam, RFLs usually use a passive single-mode fiber as the Raman gain medium. The use of single-mode fiber, however, limits the efficiency of pump coupling to the fiber, especially when a multimode pump beam is used; hence the overall conversion efficiency suffers. A first approach to solve this problem is to use a double-clad passive fiber to allow multimode cladding pumping in a Raman fiber laser (Codemard et al. 2006). It takes advantage of low threshold and a diffraction-limited output in a single-mode core. However, the use of a single-mode core may not be desirable if power scaling

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is a goal due to optical damage limits. In addition, double-clad passive fibers have high attenuation losses and requires special fabrication technologies. An alternative approach is to use multimode GRIN fibers, which permit a higher coupling efficiency and better overlap between the pump and Stokes beams in the fiber. Both factors should lead to higher conversion efficiency of the Raman laser. Since multimode GRIN fibers tend to support a fundamental fiber mode with a larger mode area, it is reasonable to expect higher-output powers. An additional advantage of using standard commercial GRIN fibers is that they are well engineered and have undergone technological maturation due to their telecom use, so that the background loss is quite low. Since the Raman laser is based on the pump-induced Raman gain in a passive fiber, it has a fundamental difference in lasing properties as compared with Er- or Yb-doped fiber lasers, namely, it features a small quantum defect, fast response of the gain on pump variations, low background spontaneous emission, as well as the absence of photo-darkening effect that is a problem in doped active fibers, especially at short wavelengths. Baek et al. first demonstrated a nearly single-mode operation of a RFL that uses a multimode fiber as the Raman gain medium (Baek and Roh 2004). The all-fiber laser cavity includes a couple of multimode fiber Bragg gratings and commercial GRIN multimode fiber with 50-m core diameter (Fig. 12). The fiber length of 40 m was chosen as a compromise between two requirements: ensuring a sufficient gain length for beam cleanup and minimizing the attenuation in the fiber. RFL was pumped by multimode Nd:YAG laser at 1.064 m with beam quality parameter M2  7. The beam quality of the Stokes wave was determined by measuring of the spot size variation of the beam in the vicinity of the focal point of a lens with a frame grabber system. It is M2 D 1:66 at 0.8 W output power. The slope efficiency of the first RFL based on multimode GRIN fiber was only 7%, but further optimization of the cavity resulted in enhancement of the laser efficiency and the output power (Terry et al. 2007b). In the experiment, two

50 µm core Multimode Fiber

Optical Spectrum Analyzer Diffraction Grating & Powermeter Beam Profile Analyzer

5X

10X

Fiber Grating

Fiber Grating

Diode-Pumped nd : YAG

Fig. 12 Schematic diagram of the multimode RFL (Baek and Roh 2004)

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Fig. 13 Beam quality of the RFL with various output FBGs for two different Stokes output powers (Terry et al. 2007b)

temporally incoherent CW Nd:YAG lasers were combined using polarizing beam splitters and then launched into the RFL based on 2.5-km multimode GRIN fiber with 50 m core. The cavity of the RFL was almost the same, as shown in Fig. 12. The input FBG possessed a single-mode reflectivity of 99% at the first Stokes wavelength. The peak single-mode reflectivity of the output FBG varied from 30% to 99% or it was replaced by 4% Fresnel reflectivity of the normally cleaved fiber end. It was shown that the Fresnel reflection provided higher slope efficiency of the RFL, and the M2 value appeared to increase with the Stokes power and reflectivity of the output FBG (Fig. 13). It is also promising to pump the multimode RFL directly with multimode diodes. This approach eliminates intermediate conversion stage and benefits from the wide wavelength coverage available from high-power laser diodes (800–980 nm). In this way it is possible to generate Stokes wavelengths which are not covered by alternative practical fiber laser sources, e.g., in 830–1000 nm range. Note that power of LDs operating at 915 and 980 nm has already exceeded the 100 W level for a single unit.

GRIN Fiber Raman Lasers Directly Pumped by Multimode Laser Diodes This concept was developed in two directions. The first one includes RFL cavity based on bulk optical elements (Yao et al. 2015; Glick et al. 2016a, 2017). The second one follows Baek and Roh (2004) and Terry et al. (2007b) in developing an all-fiber RFL cavity by means of FBGs (Kablukov et al. 2013; Zlobina et al. 2016; Zlobina et al. 2017a) and also focuses on the all-fiber coupling of pump radiation into the laser cavity (Zlobina et al. 2017b). The last approach offers a simple and reliable all-fiber configuration of high-power RFL with good beam quality. The typical RFL configuration with bulk-optic cavity is shown in Fig. 14 (Yao et al. 2015). Beams of two high-power multimode LDs operating at 975 nm are

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f = 20 mm HR f = 8 mm

f = 11 mm VBG

DM 1 DM 2

f = 20 mm

Laser output

Power meter & OSA

Normalised refractive index (a.u.)

Graded-index multimode fiber 1.2 0.9 0.6 0.3 0.0 –80 –40 0 40 80 Transverse position (µm)

Fig. 14 Experimental setup of the CW RFL with GRIN Raman gain fiber (Yao et al. 2015)

combined into a single beam in an angled narrowband volume Bragg grating. The wavelength-combined pump light is launched into a 62.5 m multimode GRIN fiber via a lens. Dichroic mirror placed in the far end of the fiber was used to launch pump and Stokes light back into the fiber. It has reduced reflectivity at the second Stokes wavelength to prevent cascaded Raman generation. The left side of the fiber was normally cleaved and acted as a 4% output coupler of the cavity. The slope efficiency of such RFL reached 81% in a 1.5-km-long GRIN fiber and output power amounted to 19 W. The measured beam quality of the generated beam at 1019 nm equals to M2 D 5, while it was 22 for the input pump beam. Yao et al. (2015) achieved a good agreement of the experimental and calculated power characteristics when assumed that the pump was uniformly distributed over the 1530 m2 effective area which is two times less than the core area of 3070 m2 . Therefore, the pump intensity was more likely concentrated near the center of the core. When the GRIN fiber was replaced by 650 m special double-clad fiber, the output beam quality was improved to M2 D 1:9, but the output power was reduced to 6 W because of comparatively high propagation loss of the specialty double-clad fiber. Glick et al. (2016a) used a basic configuration which was similar to that shown in Fig. 14. Output Stokes power as high as 85 W was generated at 1020 nm when they used a 0.5 km GRIN fiber and 978-nm pump LD which was operating in a quasi-CW regime of 20 ms duration and 15 Hz repetition rate. The slope efficiency of the RFL equals to 70%. M2 values for the Raman output beam were measured to be from 2.9 to 5.6, while it amounted to 12–14 for the input pump beam. In general, the Stokes beam quality deteriorates at higher-power levels, repeating the behavior of the incident pump beam (Fig. 15a). The reduction of the laser beam quality was explained as follows. When the pump power increases, the Raman lasing threshold is reached in more peripheral regions of the fiber core, so those areas also begin to participate in Raman conversion, and the effective Stokes beam diameter becomes larger, thus resulting in poorer beam quality. The brightness enhancement is determined as the ratio of the Raman output brightness to the pump brightness,

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Fig. 15 (a) M2 of pump and Raman beams and brightness enhancement (BE) versus launched pump power. Open circles are M2 of pump before entering fiber, and open squares are M2 of pump after propagating through fiber. Triangles are M2 of Raman output. Solid circles are BE of Raman compared to pump before fiber and solid squares are BE of Raman compared to pump after fiber (Glick et al. 2016a); (b) Raman output power and efficiency as a function of launched pump power in 0.2-km-long RFL (Glick et al. 2017)

. 2 defined as B D P M2  , is also shown in Fig. 15a. Comparing the Raman output to the pump after its double propagation through the fiber, a maximum brightness enhancement of 7.3 was obtained which is comparable to factor of 5.2 reported in Yao et al. (2015). Further power scaling was performed by adding the second 978-nm LD in the scheme and shortening the GRIN fiber to 0.2 km (Glick et al. 2017). One advantage of the fiber shortening was the higher threshold for the second Stokes wave. The other advantage was the lower passive losses. Reduction of fiber length resulted in an increase of the laser threshold by a factor of 2.5 compared to Glick et al. (2016a). However, the output power and slope efficiency in CW regime were also increased to 154 W and 90% correspondingly (Fig. 15b). The M2 value of the Raman output was measured to be between 4 and 8 at increasing the output power from 20 to 154 W, while the pump beam before entering the fiber showed M2 D 16–18. The maximum brightness enhancement 8.4 almost did not change with the fiber length shortening. Despite the outstanding results in output power and slope efficiency obtained in Yao et al. (2015) and Glick et al. (2016a, 2017), the generated wavelength of 1020 nm was not as interesting because a much higher power at this wavelength is available from YDFLs (Glick et al. 2016b). Kablukov et al. (2013) earlier reported on the first CW RFL operating below 1 m. Pump radiation from 938 nm LD was launched via collimating lens into 62.5 m GRIN fiber with an in-fiber cavity formed by a highly reflective FBG and 4% Fresnel reflection from the cleaved fiber end (Fig. 16a). The RFL threshold power was estimated as:

Pth D

2˛s L  ln .R1 R2 / 2gR Leff

 ı   where Leff D 1  exp L˛p ˛p is the effective absorption length.

(22)

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Fig. 16 (a) Experimental setup of the CW RFL with GRIN Raman gain fiber, FBG reflector (R1) and Fresnel output (R2). (b) Calculated threshold power versus fiber length in the cavity formed by mirrors with R1 D 90% and R2 D 4% (Kablukov et al. 2013)

Following the results of calculations, the threshold power Pth takes a minimum value in the range of cavity lengths between 2.5 and 4.5 km (Fig. 16b). According to preliminary calculations, the generation threshold at the wavelength of 980 nm in the experiment with 4.5-km-long GRIN fiber was around 35 W. The forward Stokes power grows to 2.3 W when threshold of the second Stokes wave at 1025 nm is achieved. The RFL slope efficiency of 25% was not high. The low value of the second threshold was explained by the low-index transverse modes generation. The generated spectrum included three individual peaks (Fig. 17a), while the main part of the power was concentrated in the peak at 980 nm corresponding to the fundamental mode. The far-field profile of the output beam demonstrated a reduction of the beam divergence by three times compared to the pump beam (Fig. 17b). Use of an additional mirror at the fiber end which reflected residual pump power at 938 nm back to the cavity led to the increase of the output power and slope efficiency up to 3.3 W and 35%, respectively. Babin et al. (2013) reported on endeavors to get Raman lasing in a LD-pumped passive fiber without a conventional resonator, using a positive feedback provided by a randomly distributed Rayleigh backscattering. Though the Rayleigh backscattering integral reflection coefficient is low (0.1%), the scattered radiation is amplified after random reflections so that the integral Raman amplification may compensate the losses thus reaching laser threshold even in the absence of conventional cavity mirrors, similar to random lasing in single-mode fiber Raman lasers (Turitsyn et al. 2010). The experimental study of random lasing is performed on the basis of the RFL scheme with direct LD pumping (Fig. 16a), in which the narrowband FBG has been replaced by a broadband mirror. Moreover, instead of the 4% Fresnel end reflection (removed by cleaving the fiber with angle >15o against a normal one), a distributed Rayleigh backscattering was employed for feedback. The random distributed feedback laser threshold increased to 42 W. The generated Stokes radiation at 980 nm reached 0.3 or 0.5 W in the RFL configuration with or without an additional mirror reflecting back the residual pump power. The relative reduction of the generated beam divergence was by 4.5 times as compared with that for the pump beam, i.e., higher than in the case of SRS beam cleanup only (Kablukov et al. 2013). Therefore, an additional influence of the SBS (Rodgers et al. 1999) present near threshold and the Rayleigh backscattering distributed feedback make the cleanup effect stronger. Due to the high beam quality of the generated Stokes wave, such a laser has a very low threshold of the second-order Stokes wave generation.

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a

b

Fig. 17 (a) Output spectrum of the generated Stokes wave. (b) Output beam profile for the pump (black) and Raman (gray) lasers (Kablukov et al. 2013)

Combined Action of Raman Beam Cleanup and Mode-Selecting FBGs in GRIN Fiber Raman Lasers A further increase of the generated power in GRIN fiber-based RFLs with FBGs was achieved by optimization of the cavity length, which was obviously larger than optimal, as the transmitted pump power was relatively low in Kablukov et al. (2013) (10% of the coupled pump power). Optimization of the GRIN fiber length and utilization of the 915 nm pumping allowed Zlobina et al. (2016) to increase the output power and slope efficiency while decreasing the laser wavelength down to 954 nm. The experimental scheme is similar to that in Fig. 16a. The linear cavity of the laser was formed by high-reflection fiber Bragg grating FBG1 and normally cleaved fiber end providing weak (4%) Fresnel feedback, or output fiber Bragg

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grating FBG2 with low reflection. The multimode FBGs were written by a highpower UV argon laser in the same multimode GRIN fiber. The experimental power threshold of 30 W is nearly the same for two fiber lengths (L D 3:7 and 2.5 km), in correspondence with the calculated threshold power curve (see Fig. 16b). At the same time, the differential efficiency is higher for the shorter length amounting to 37% and 26% at L D 2:5 and 3.7 km, respectively. Optimization of highly reflective grating FBG1 resulted in increasing the generated power to 4.7 W and the slope efficiency to 41%. Since the cascaded generation of higher Stokes orders limits the power for the first Stokes, it is possible to increase the second Stokes threshold by substituting the Fresnel reflection by low reflective (3%) output FBG2 at 954 nm. The second Stokes wave was not observed in this case, but the laser threshold and generation power worsen in comparison with the normally cleaved output end. Figure 18a shows the output Stokes spectrum measured for the scheme with (solid line) and without (dash-dot line) output FBG2. The spectrum is smooth without FBG2. Its 3 dB linewidth of 0.7 nm is defined by the FBG1 bandwidth.

Fig. 18 Output spectra of the Stokes wave for L D 2:5 km RFL: (a) with normally cleaved fiber end (dash-dot line) or with the FBG2 (solid line); (b) at different temperatures of the FBG2 (Zlobina et al. 2016)

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With the FBG2 the spectrum has multi-peak structure which is similar to one obtained with another multimode FBGs (Terry et al. 2007b; Kablukov et al. 2013) and is mainly defined by the FBG2. The wavelength difference  between two neighboring groups of the transverse modes in GRIN fiber (Gloge et al. 1973; Kawakami and Tanji 1983) can be estimated using Eq. (23) for propagation constant ˇ from Mizunami et al. (2000) and condition ˇ D = for reflection at FBG with period :    D 2 NA= d n2cl ;

(23)

where  is wavelength of the fundamental mode, NA is numerical aperture, d is the core diameter, and ncl is the cladding refractive index. Estimated value of the wavelength difference  D 0:61 nm is in a good agreement with the distance between two lines 0.62 nm experimentally observed in the case of FBG2 (see Fig. 18a). Therefore those lines correspond to two neighboring groups of low-index transverse modes which are reflected by FBG2. Heating of the FBG2 to 74 degrees resulted in synchronous shift of the two-peak structure by 0.6 nm (see Fig. 18b). As a next step, the possibility of mode selection in graded-index fiber Raman laser by means of FBG mode-selection properties was studied in Zlobina et al. (2017a). To improve the selection of the fundamental mode of graded-index fiber, instead of conventional UV FBG2 (recorded by CW UV radiation), a FS FBG2 was spliced at the laser output. It was point-by-point recorded in the central region of the fiber core by a femtosecond (FS) laser beam (Dostovalov et al. 2016). Reflection spectra of UV FBG1 and FS FBG2 are shown in Fig. 19a. The FS FBG2 and UV FBG2 were first compared in a 2.5-km-long Raman laser (Kablukov et al. 2016) (see Fig. 19b). The measurements revealed the correspondence between the FBG reflection spectra and the generation spectra (Fig. 19). When using the highly reflective UV FBG1 with nonselective Fresnel reflection (4%), one can observe the generation of a relatively homogeneous spectrum of 1 nm width (dashed curve), because of the relatively wide spectrum of UV FBG1 where individual resonances cannot be resolved. In the case of output UV FBG2 (which was also used in Zlobina et al. 2016), one can observe a three-peak structure with peaks spaced by 0.6 nm (see Eq. (23)), which corresponds to the reflection of three individual groups of graded-index fiber modes with small transverse indices (dotdashed curve). The reflection spectrum is narrow for each group of modes; therefore, the generation spectra of different mode groups are well resolved. In the case of output FS FBG2, one observes a two-peak generation spectrum with a distance of 1.2 nm between the peaks (solid curve). The doubling of the distance between the peaks can be explained by the absence of the reflection peak for the second-group mode (see Fig. 19a) (because of the small overlap integral for the field of this mode with the grating recorded in the central part of the fiber core). Therefore the second group of modes (at a wavelength of 953.6 nm) is not involved in lasing. The peak of the third group at 953 nm is unstable in time and manifest itself only in the 2.5-kmlong laser. Stable single-mode lasing was achieved in shorter GRIN fiber.

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Fig. 19 (a) Reflection spectra of highly reflective UV FBG1 (dashes) and output FS FBG2 (Zlobina et al. 2017a). (b) Generation spectra at 1.9 W output power for 2.5-km GRIN RFL with different output coupler: 4% cleaved end (dash), UV FBG2 (dash-dot), and FS FBG2 (solid line) (Kablukov et al. 2016)

It is known (see, e.g., Mafi 2012) that the mode-group number in a gradedindex fiber can be characterized by the quantity g D 2p C jmj  1, where p and m are, respectively, the radial and azimuthal mode numbers. The radial (r) and azimuthal ( ) field distributions for modes LPmp are described by the Laguerre polynomials (21) Lm p . Hence, the first two mode groups are not degenerate and contain modes LP01 and LP11 , while the third group includes two modes: LP02 and LP21 . The fundamental-mode diameter in the graded-index Corning 62.5/125 fiber can be estimated as 9.8 m (Mafi 2012), and the photomodification region in FS FBG2 has the form of an ellipse in the fiber cross section, with characteristic sizes of principal axes of about 1 and 10 m (see in Fig. 20a). Despite the comparable values of the mode diameter and the photomodification region size in one of the directions, the overlap integrals are small for modes with nonzero azimuthal index m. The modes with index m D 0 are maximally overlapped with the FBG recorded near

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Fig. 20 (a) Microscope image of FS modification area in the core of GRIN fiber for FS FBG2. (b) Far-field image of the output beam near the Raman threshold in the RFL configuration with FS FBG2. Pump beam is gray, Stokes beam is bright. (c) M2 measurement for residual pump radiation (triangles), Stokes generation in RFL with Fresnel reflection (squares) and FS FBG2 (circles) (Zlobina et al. 2017a)

the fiber center. Therefore, only the reflection peaks corresponding to these modes (LP01 in the first group and LP02 in the third group) arise in the reflection spectrum of the FS FBG2 and, correspondingly, in the generation spectrum of the Raman laser when this grating is installed at the output. Shortening the GRIN fiber to 1.1 km results in the selection of a single transverse mode by means of FS FBG 2; see Fig. 20b, c (Zlobina et al. 2017a). Independent measurements of the output beam in the laser with a 1.1-km-long cavity revealed its quality factor M2 to be better than 1.2 at output powers in the range from 5 to 10 W. This fact indicates that the lasing is close to single-mode. Power characteristics of 1.1-km single-mode RFL based on 62.5 m GRIN fiber with bulk

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optics (Zlobina et al. 2017a) were compared with that for all-fiber coupling of the pump radiation (see Fig. 21 Zlobina et al. 2017b), in which the output radiation of three high-power multimode LDs at wavelength of 915 nm is combined by a 3  1 multimode fiber pump combiner with an output port made of 100-m MM GRIN fiber. It is fusion spliced to the RFL cavity based on GRIN fiber with 85-m or 62.5m core. The laser cavity was formed by highly reflective (UV FBG1) and output (FS FBG2) gratings inscribed in the core of the same graded-index fiber. The obtained power and spectra in two configurations with 62.5-m MM GRIN fiber are compared in Fig. 22. The laser generation threshold increases from 40 to 50 W when changed from the bulk-optic pumping to the all-fiber one. The slope efficiency increases from 38% to 47% (Fig. 22a). Increase of the second threshold power from 10 to 16 W of generated power results from the elimination of Fresnel reflection at the input fiber facet after the modification from free-space to all-fiber

Fig. 21 Experimental setup of the all-fiber CW RFL with GRIN Raman gain fiber, in-fiber FBG reflectors and fiber pump combiner (Zlobina et al. 2017b)

Fig. 22 (a) Power of residual pump (squares) and generated Stokes (circles) waves as a function of launched pump power for the bulk-optic (empty symbols) (Zlobina et al. 2017a) and allfiber (filled symbols) pump coupling to the same 62.5-m GRIN fiber with FBGs. The second Stokes thresholds are shown by arrows. (b) Corresponding generation spectra at the second Stokes threshold for the all-fiber and bulk-optic pump coupling configurations (Zlobina et al. 2017b)

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pump coupling. As a result, the maximum power at 954 nm increases to about 16 W at 82 W launched pump power, whereas the generation linewidth (at 3 dB level) is left at the level of 0.4 nm (Fig. 22b). At the same time, after the transition to all-fiber configuration, the RFL operation becomes much more stable. The combined pump power of three 915-nm LDs is sufficient to observe Raman lasing at 954 nm in 85-m GRIN fiber, for which the threshold is 85 W (Fig. 23a). The second Stokes threshold grows from 135 to 180 W of coupled pump power at reducing fiber length from 1.95 to 1.5 km, whereas the first Stokes threshold remains nearly the same. As a result, maximum power at 954 nm grows from 21 to 49 W, while the residual pump power varies from 20 to 25 W only. The threshold grows by almost two times in proportion with the GRIN fiber core area increase. However, the slope efficiency of pump-to-Stokes conversion is higher for the 85-m GRIN fiber (67%) than that for 62.5-m one (47%) in spite of sufficiently stronger integral pump attenuation. It may be caused by a lower quality of the generated beam in this case. To check this, the output beam profile was measured (Fig. 23b) as well as beam quality factor M2 D 2:6 at 32 W output power. It is almost two times of the quality factor for the 62.5-m fiber laser (M2  1:3), while the beam profile appears to be close to Gaussian. The measured M2 value is only slightly ( of the output beam on the fiber modes where calculated along the propagation coordinate z at t D 0. In Fig. the relative mode occupation jap;q .z/j2 =N .z/ P 26 we illustrate 2 (where N .z/ D p;q jap;q .z/j / after 0.2 mm of propagation only, for a low-power pulse (panel a) and a high-power pulse (panel b), respectively. Panels c and d show the corresponding instantaneous (at t D 0) relative intensity profiles of the beams. Figure 27 shows the same decompositions but obtained after 1 m of propagation. As can be seen, nonlinearity (panels b and d) substantially modifies the modal distribution of the beam, by increasing the mode H0;0 .x; y/ content, in agreement with the hypothesis of self-cleaning toward the fundamental mode of the GRIN MMF.

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Fig. 27 Same as in Fig. 26, with zD1 m (Reproduced from Krupa et al. 2017b)

Kerr Beam Cleanup in GRIN MMF Let us start the description of Kerr-induced spatial beam self-cleaning experiments by considering a lossless propagation environment, involving just a few meters of standard commercially available GRIN MMF. Initial experiments were carried out with 30–900 ps duration input pulses, propagating in the normal dispersion regime (Krupa et al. 2016b, 2017a, b; Wright et al. 2016). Subsequently, Kerr self-cleaning was also investigated in the femtosecond pulse regime, where the power threshold grows larger and it becomes nearly comparable with the threshold for catastrophic self-focusing (Liu et al. 2016). Figure 28 shows details of the observed power dependence of Kerr beam cleanup in a 12m-long GRIN MMF with 52 m of core diameter. In these experiments, 900 ps pulses with 30 kHz repetition rate from an amplified Nd:YAG microchip laser operating at 1064 nm were used (Krupa et al. 2017a, b). Panels (a–d) of Fig. 28 show the 2D near-field spatial beam pattern emerging from the fiber at 1064 nm, as a function of the output peak power (Ppp ). Panels (a’–d’) of Fig. 28 display the corresponding beam transverse profiles. At low powers (Ppp D 3:7 W), where propagation is linear, the highly multimode excitation including about 200 guided modes leads to a highly irregular speckled beam distribution (Fig. 28, panels a, a’). However, when progressively increasing the input pulse power, the output beam

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Fig. 28 Experimental spatiotemporal nonlinear dynamics in GRIN MMF. (a–d), Near-field images of the MMF output at the pump wavelength of 1064 nm (intensities are referred to the local maximum and are shown in linear scale; scale bar: 10 m) showing spatial beam selfcleaning toward a well-defined bell-shaped distribution when increasing the output peak power Ppp . (a’–d’), Corresponding beam profiles versus x (yD0 section). (e), Corresponding output spectra. Fiber length: 12 m (Reproduced from Krupa et al. 2017a)

profile evolves into a well-defined bell-shaped spot at the beam center, whose diameter is close to the fundamental mode of the fiber, on a residual low-power background (Fig. 28, panels d, d’). Spectra in panel e of Fig. 28 reveal that the redistribution of power toward a central quasi-Gaussian profile results before any substantial spectral broadening occurs (Krupa et al. 2017a, b). Kerr beam cleanup leads to a significant increase of the spatial quality of output beam and its brightness: a nearly threefold decrease of the output near-field beam FWHMI (full width at half maximum intensity) diameter was measured. A statistical study (involving a large number of input beams, characterized by different orientations of their linear state of polarization) confirmed that Kerr beam selfcleaning is associated with a reduction in the average beam diameter, but most importantly with a dramatic reduction in the standard deviation of the output beam size (Krupa et al. 2017a, b). It is also important to mention that, for powers above the Kerr-induced spatial beam self-cleaning threshold, one observes significant spectral and temporal reshaping of the input pulses, leading to sideband series and supercontinuum generation (Wright et al. 2016; Krupa et al. 2016a, b; LopezGalmiche et al. 2016).

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Fig. 29 Experimental evolution of near-field spatial pattern of GRIN MMF output at 1064 nm as a function of propagation distance z (a–f) induced by Kerr beam self-cleaning; High input power D 68 kW; Low input power D 0.007 kW; Scale bar: 10 m (Readapted from Krupa et al. 2017a)

To gain additional insight into the beam self-cleaning process, cutback measurements as in Fig. 29 were performed. These results confirm that spatial reshaping occurs after a certain distance of propagation (here, this is only about 30 cm, owing to the relatively high input pulse powers (68 kW)). These observations prove that beam self-cleanup occurs in the presence of an initial highly multimode excitation. These observations are in quite good qualitative agreement with the numerical simulations described in the previous subsection of this chapter. To further analyze the possible interplay between spatial and spectral degrees of freedom of the light pulses in the process of beam cleaning, one may image the fiber output face on a CCD camera through a dispersive spectroscope. Note that a standard spectrum analyzer cannot resolve the cavity mode spacing of a microchip laser. The imaging spectroscope was based on a heavily dispersive grating in a Littrow configuration. With such setup, one obtains an image that combines the transverse spatial pattern (vertical coordinate) and the frequency spectrum (horizontal coordinate). Spectral broadening leads to an image expanding along the horizontal axis. Spatial beam narrowing or broadening is instead visible in the vertical axis. Figure 30 shows three representative sample images. Panel (a) of Fig. 30 was recorded at low powers and shows that the output spectrum is characterized by a nearly single longitudinal mode of the input laser, plus a weak satellite longitudinal

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Fig. 30 Spatial (x) and spectral () analysis of output wave for three pump powers: (a) linear propagation 0.2 kW, (b) high power 7 kW, (c) very high power 55 kW; pulse duration: 0.9 ns, fiber length: 3 m (Reproduced from Krupa et al. 2017b)

mode. In the case of linear propagation, one obtains an output beam with an extended speckled spatial pattern (30 m in the vertical dimension). Panel (b) of Fig. 30 shows that in the nonlinear propagation regime, beam self-cleaning leads to beam shrinking to about 10 m in the vertical direction, with no discernible modification in the frequency spectrum. For much higher pump power values, panel (c) in Fig. 30 shows that, along with spatial compression, substantial spectral broadening appears. In these experiments we used a 3-m-long fiber, instead of the 12-m-long fiber as in Fig. 28. The difference in fiber lengths explains the higher level of power required for self-cleaning. Similar results are obtained with different powers and fiber lengths, provided that their product remains unchanged, similarly to the Kerr effect in single-mode fibers. An important quantitative measure of the beam quality at the output of the GRIN MMF is given by its M 2 parameter. The dependence of M 2 as a function of the input beam power is illustrated in Fig. 31. Here the M 2 parameter was calculated as a second moment width function, which is more appropriate for speckled beams. Figure 31 reveals that at about 10 kW (for a fiber segment of 3 m), the beam quality reaches a stationary value close to 4: this value is then kept nearly constant even for input peak powers much above the threshold for self-cleaning. The images in the insets of Fig. 31 show corresponding transverse beam profiles. Although numerical simulations based on the solution of Eq. 24 are indeed capable of predicting Kerr beam self-cleaning, a simple physical explanation of this effect is still missing. In fact, a few years ago Picozzi et al. have predicted, with the help of a statistical approach, the condensation into the fundamental mode (surrounded by a background equipartition of energy into the higher-order modes) of an initial highly multimode beam in the frame of the GRIN fiber model of Eq. 24 (Aschieri et al. 2011). This condensation of classical waves is associated with the generation of a Rayleigh-Jeans equilibrium distribution. This interpretation is fascinating; however it is associated with some predictions that are not yet fully corroborated by the experiments. For example, the wave condensation model

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Fig. 31 M 2 measurements at the output of a 3-m-long fiber, and for the X- and Y-axis (Reproduced from Krupa et al. 2017b)

predicts that a faster convergence toward the condensed state occurs whenever the input beam is equally distributed among all modes, whereas experiments seem to indicate that Kerr beam cleanup into the fundamental mode requires an initial unbalance toward this mode in the input beam decomposition which is coupled to the MMF. Another interpretation of Kerr beam self-cleaning has been put forward by L. Wright et al., who have conjectured that irreversible evolution toward the fundamental mode is achieved since this mode is the most unstable with respect to nonlinearity induced perturbations (Wright et al. 2016). Although this interpretation may look as a paradox, it finds its justification when considered in the broader framework of the theory of self-organized criticality, which is a universal mechanism for pattern formation and self-organization in out-of-equilibrium physical systems. Finally, a perhaps more intuitive explanation for the nonlinear asymmetry of power flow toward the fundamental mode of the GRIN MMF may be based on the nonlinear nonreciprocity of the mode-coupling process in a GRIN MMF. The periodic oscillations of the multimode beam intensity, which occur along the fiber through mode beating, lead to Kerr effect induced longitudinal modulation of the fiber refractive index or dynamic long-period fiber grating. This grating may provide quasi-phase-matching for a variety of nonlinear four wave-mixing processes, leading to energy exchange between the fundamental and the HOMs. A simplified explanation of the physical mechanism behind the Kerr-induced spatial beam self-cleaning effect has been put forward, based on a two-mode mean-field model; see for details Krupa et al. (2017a, b). This mean-field model predicts that, for powers above a certain threshold power value, the reciprocity of mode coupling is broken, thus leading to a preferential flow of energy toward the fundamental mode.

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Kerr Beam Cleanup in Step-Index Active MMF with Loss or Gain When a few meters of MMF are used, experiments described in the previous subsection demonstrate Kerr beam cleanup in a GRIN fiber. However, similar experiments performed so far with a step-index MMF could not demonstrate selfcleaning, at least until power levels such that the fiber damaging threshold was approached. Now, an interesting question is if some modification to the linear fiber parameters could be introduced that could allow for the observation of beam self-cleaning with a step-index MMF. Again quite surprisingly and very interestingly, experiments have shown that this feature is possible, whenever the linear propagation environment is strongly dissipative, that is, in the presence of strong linear loss of gain. This is quite counterintuitive, as one would normally imagine that the delicate balance between the Kerr-induced long-period fiber grating and resulting power conversion into the fundamental mode could be disrupted in the presence of strong dissipation. To verify such a possibility, Kerr beam self-cleaning was experimentally studied in a MMF doped by rare-earth atoms, by using the setup shown in Fig. 32. Two different configurations were employed. In the first configuration, 3-m-long, highly lossy (2.44 dB/m at 1064 nm), unpumped double-clad ytterbium-doped MMF with a nearly step-index profile was used (see Fig. 33). In the second configuration, the same Yb-doped fiber was pumped by a CW laser diode, which provided up to GD20 of gain. The Yb-doped MMF had a 55 m core diameter, and it was surrounded by a D-shaped inner cladding of 340  400 m size, for guiding the pump radiation (Guenard et al. 2017a). In spite of the large overall attenuation of about 7.3 dB and the non-parabolic refractive index profile, experiments with the unpumped Yb-doped MMF still led to the observation of the Kerr beam cleanup. Figure 34 shows that, in the lossy case, the M 2 parameter of the output beam dropped from 16 (in the low-power regime) down to a value of only 2 after beam cleanup. Let us note however that the nonlinear beam reshaping process was qualitatively different from that in the GRIN MMF. In fact, the input pump power was not only coupled into the Yb-doped MMF core but also into HOMs leaking into the D-shaped cladding. At high pump powers, beam

Fig. 32 Experimental setup to observe Kerr self-cleaning in Yb-doped fiber

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Fig. 33 Image (SEM) (a) of the double-clad ytterbium-doped multimode optical fiber and (b) its core refractive index profile (Reproduced from Guenard et al. 2017a)

Fig. 34 M 2 measurement (1=e 2 diameter) of the output beam pattern versus the input peak power with the passive (unpumped) Yb-doped MMF; Insets: output beam patterns for (a) low power and (b) high power. The upper scale in red gives the path-averaged power < P >z to include the impact of the fiber losses. The dashed line is an exponential fit as a guide line for the eye (Reproduced from Guenard et al. 2017a)

cleaning into the fundamental mode of the core was accompanied by a significant reduction (by about 50%) of the energy carried by the leaky HOMs. More interesting for the potential application to high-power MMF-based amplifiers and lasers, it is the possibility to obtain beam cleanup (with a much reduced power threshold) in the active configuration involving the diode pumped Yb-doped

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Fig. 35 M 2 measurement (1=e 2 diameter) of the output beam pattern versus gain of the amplified Yb-doped MMF; Input signal peak power: 0.5 kW; Insets: output beam patterns for low amplification (a) and high amplification (b) (Reproduced from Guenard et al. 2017a)

MMF. To prove this issue, in experiments, the power of the microchip Nd:YAG laser was kept fixed at 0.5 kW, and the D-shaped inner cladding of the Yb-doped MMF was pumped with a CW laser diode at 940 nm with increasing values of its power level. The insets in Fig. 35 depict the evolution of the near-field pattern at the fiber output, as a function of optical gain (G) of the Yb-doped MMF. Once again, Kerr beam cleanup was observed. The decrease of the M 2 parameter presented in Fig. 35 further confirms the substantial improvement of the output beam quality, from a value of 9 with GD5 down to only 2 in the self-cleaning regime (for a gain GD20 or larger). Let us recall that an input peak power as high as 40 kW is necessary with the same lossy Yb-doped MMF, whereas a similar length of GRIN MMF exhibits a threshold peak power of about 7 kW. Figure 35 shows the evolution of the M 2 upon the gain of the amplified Yb-doped MMF, and Fig. 36 provides an interesting comparison of the longitudinal power evolution in the lossy (unpumped) and in the active (pumped) Yb-doped MMF, respectively. In particular Fig. 36 reveals that, if we consider the values of the output peak power instead of the input values, both cases lead to nearly the same value of about 11 kW, which, incidentally, is also the typical output power value that is obtained for a lossless GRIN MMF of the same length. Moreover, Fig. 36 also

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Fig. 36 Comparison of self-cleaning in the lossy and the active Yb-doped MMF. Here the power evolution along the fiber is compared in two situations leading to Kerr beam cleanup: remarkably, the output pulse power is nearly equal in both cases

reveals that the path-averaged power (< P >z ) is significantly reduced (by about six times) by the presence of gain with respect to the case with loss. Therefore, in the nonlinear regime the presence of loss and gain cannot be interchanged, by simply considering the value of the path-average power.

Self-Cleaning in a MMF Laser Cavity Similar to the case of Raman beam cleanup in GRIN MMF that was described in the previous section, Kerr beam cleanup can also be usefully exploited to operate a MMF laser with a high-power, quasi-single-mode transverse output beam profile. The key difference here is that the Kerr beam cleanup occurs at the pump oscillation frequency and not at the Stokes-shifted frequency. To compensate for cavity loss, in the case of a Kerr beam self-cleaning-based MMF laser, it will be necessary to provide gain via the use of an active MMF such as the Yb-doped MMF that was described in the previous subsection. As a matter of fact, lasers based on MMFs could provide an useful alternative to large mode area (LMA) single-mode fibers for the generation of high peak power pulses and for fabricating high-power fiber amplifiers and lasers. This is mainly because of cost and modal instability issues of lasers based on LMA fibers. Kerr beam cleanup could then remove the main obstacle which has limited the use of MMFs for fiber lasers so far that is their inherent degradation of the spatial beam quality.

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Fig. 37 Experimental setup to observe Kerr self-cleaning in composite laser cavity including Ybdoped fiber

In this section, we briefly describe the recent demonstration of Kerr beam cleanup in a coupled cavity laser (Guenard et al. 2017b), composed by a passively Q-switched Nd:YAG microchip laser plus an external cavity including the same Yb-doped MMF amplifier as presented in the previous subsection (see Fig. 37), and a polarization control (including two wave plates and a polarizer) that permits to adjust the level of coupling among the two cavities and the power injected into the MMF. Such type of coupled cavity lasers, but with single-mode fibers in the external cavity, was studied before for timing jitter reduction in Q-switched laser and supercontinuum generation (Bassri et al. 2012). Using a MMF in the external cavity, one may simultaneously shape the output beam in the spatial, spectral, and temporal domains. We obtained both Q-switched and Q-switched mode-locked operations, in combination with Kerr beam cleanup in the MMF Yb-doped fiber. In order to form the external cavity, a mirror was inserted after the Yb-doped MMF, as shown in Fig. 37. Owing to the large length difference between the microchip cavity (7 mm) and the MMF cavity (1 m), it is always possible to find common resonance frequencies for the two cavities, without an electrooptic servo control. Let us consider first the case of an unpumped, lossy Yb-doped MMF in the external cavity. Whenever relatively low-power pulses are launched in the MMF from the microchip laser, the two cavities are weakly coupled, and a highly multimode beam recirculates in the MMF. When cavity coupling is sufficiently increased by means of the polarization control, a sudden increase of power in the MMF section occurs. This leads to the excitation of multiple longitudinal modes in the laser spectrum (see the upper right panel in Fig. 38) and to the switching by 90ı of the polarization state of the microchip laser. Simultaneously, the transverse profile of the laser beam at the MMF output switches from a speckled pattern to a clean bell-shaped profile (see the upper mid panel in Fig. 38). The abrupt change in the beam profile from the MMF is associated with the improvement of the beam quality parameter M 2 dropping down to 1.7. Note that even with strong cavity coupling, Q-switched laser operation is preserved with unchanged repetition rate. However, the laser pulse is shortened from 525 to 225 ps (see the upper left panel in Fig. 38).

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Fig. 38 Temporal, spatial, and spectral characterization of output from a MMF-based Q-switched double-cavity laser, showing pulse compression, spatial beam cleanup, and spectral broadening, respectively. Top (bottom) plots: case of a lossy (pumped) Yb-doped MMF (Readapted from Guenard et al. 2017b)

Next, consider MMF laser operation when switching on the pump laser at 940 nm, in order to change the Yb-doped MMF into an amplifier. The Yb-doped MMF gain could be varied with the pump power, up to a maximum of GD23 (for an input signal of 0.5 kW peak power). Note that the available amplifier gain is limited by the damage threshold of the fiber end face. The main impact of MMF gain is the increase of the output laser power and the addition of an ASE background to the output beam transverse profile. The bell-shaped output beam obtained for maximum gain is shown in the bottom mid panel in Fig. 38. However, the beam profile is not as clean as in the passive MMF situation. The corresponding laser spectrum exhibits a discrete structure including the longitudinal modes fixed by the coupled cavities (see the bottom right panel in Fig. 38). Still, the beam quality remains high (M 2 D 1:8). The temporal profile of the output pulse acquires an asymmetric shape, with a 350 ps duration (FWHMI) (see the bottom left panel in Fig. 38). Acknowledgements The authors acknowledge financial support by the Russian Science Foundation (grant 14-22-00118) and by the Ministry of Education and Science of the Russian Federation (14.Y26.31.0017); K.K. has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. GA-2015713694 (“BECLEAN” project).

References A.B. Aceves, G.G. Luther, C. De Angelis, A.M. Rubenchik, S.K. Turitsyn, Phys. Rev. Lett. 75, 73 (1995) A.B. Aceves, O.V. Shtyrina, A.M. Rubenchik, M.P. Fedoruk, S.K. Turitsyn, Phys. Rev. A 91, 033810 (2015)

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9

Shock Waves Stefano Trillo and Matteo Conforti

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gradient Catastrophe and Classical Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regularization Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shock Formation in Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanisms of Wave-Breaking in the Normal GVD Regime . . . . . . . . . . . . . . . . . . . . . . . Shock in Multiple Four-Wave Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Focusing Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control of DSW and Hopf Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Riemann Problem and Dam Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Competing Wave-Breaking Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resonant Radiation Emitted by Dispersive Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase-Matching Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steplike Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bright Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Periodic Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shock Waves in Passive Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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S. Trillo () Department of Engineering, University of Ferrara, Ferrara, Italy e-mail: [email protected] M. Conforti CNRS, UMR 8523, PhLAM – Physique des Lasers Atomes et Molécules, University of Lille, Lille, France e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 G.-D. Peng (ed.), Handbook of Optical Fibers, https://doi.org/10.1007/978-981-10-7087-7_16

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Abstract

We discuss the physics of shock waves with special emphasis on the phenomena related to the field of nonlinear fiber optics. We first introduce the general mechanism commonly known as gradient catastrophe and the related concept of classical shock waves. Then we proceed to discuss the possible regularization mechanisms of the shock, and in particular the dispersive regularization, which is behind the formation of dispersive shock waves in fibers. We then discuss different possible scenarios that lead to observe the formation of dispersive shock waves in fibers, such as pulse propagation, four-wave mixing, and passive resonators, also showing that fibers allow for investigating the dispersive regime of classical problems related to the physics of shock such as the dam-break problem and the propagation of Riemann waves. We also discuss the phasematching mechanism that induces the shock to efficiently radiate resonant radiation in the normal dispersion regime. Throughout the text we refer to the mathematical models and the approaches that are employed to describe such phenomena.

Introduction Classical shock waves are disturbances which exhibit a steep jump of the associated physical quantities such as density, velocity, temperature, etc., which can move with their own characteristic velocity (Lax 1973; Whitham 1974; LeVeque 2004). They have been deeply investigated in connection with systems in the form of so-called conservation laws, especially in different branches of studies of fluid flows such as standard gas dynamics, hydrodynamics, or blast waves to name a few wide subareas of interest. The physical phenomenon which is behind the formation of a shock wave is the gradient catastrophe, namely, the divergence of the gradient in one point along the disturbance, which develops at a finite time (or distance, whenever the latter is used as the evolution variable instead of time). At this stage the wave is said to “break.” The essential mechanism that explains why a physical system can be driven toward the gradient catastrophe is the dependence of the velocity on the local wave elevation. This effect can drive a smooth initial disturbance to develop an infinite gradient, after which the wave disturbance tends to become multivalued. Mathematically, classical shock waves are introduced with reference to the weak formulation of the underlying conservation law to remove the multivalued stage in favor of an ideal propagating jump. In the physical reality, however, shock waves exhibit a finite extension rather than being strict local jumps, across which the physical quantities change rapidly. This is usually due to the dissipative or viscous mechanisms (neglected in the conservation laws) which, once properly accounted for, are found to regularize the abrupt changes by smoothing out the variation of the physical quantities involved in the shock. When light is considered, either a beam propagating in free space in transparent media or a pulse propagating along an optical fiber, the same type of phenomena can occur, namely, the tendency to exhibit wave-breaking, when the propagation

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occurs in the strong nonlinear regime. In this respect the light effectively behaves as a “photon fluid” showing remarkable similarities with the dynamics of gas or with the behavior of waves in shallow or deep water. However, since the viscous effects are usually negligible in this case, the shock waves turn out to be regularized by dispersive (or diffractive in space) effects. This leads to a completely different scenario, where instead of smoothing due to viscous effects, the shock features the onset of fast oscillations that spontaneously grow and expand around the steep gradient, thus regularizing the tendency of the gradient to diverge. Such type of nonstationary structures is known, nowadays, as dispersive shock waves (DSWs) (El and Hoefer 2016) or undular bores, borrowing a term which is more often used in the field of hydrodynamics (Peregrine 1966). The role of such structures and, more generally, the studies of the type of hydrodynamic behaviors where dispersive effects become prominent (dispersive hydrodynamics) have only recently become an area of active investigation also outside optics, for instance, in the areas of Bose-Einstein condensation of ultracold atoms (Hoefer et al. 2006), fluid dynamics (Maiden et al. 2016; Trillo et al. 2016), or spin waves in magnetic films (Janantha et al. 2017). Historically, in optics, such behavior has been first reported in the spatial domain with reference to the propagation of laser beams in thermal defocusing media (Akhmanov et al. 1968; Whinnery et al. 1967) and analyzed in terms of geometric optics approximation. Undulatory deformations of Gaussian modes have been interpreted as aberrations of the effective lens self-induced by the beam via heating of the medium caused by weak absorption. In this area, a much deeper connection with wave-breaking and DSW formation has been established recently (Wan et al. 2007; Ghofraniha et al. 2007; Conti et al. 2009), also remarkably extending the notion of the shock, which is intrinsically a local and ordered entity, to media with nonlocal response (Ghofraniha et al. 2007) and to disturbances constituted by superposition of incoherent waves (Garnier et al. 2013; Xu et al. 2015; Randoux et al. 2017). Since we will not discuss further these achievements here, the interested reader is referred to the original literature on these advanced topics. In time domain, shock formation and DSWs have been first realized to play a significant role in optical fibers in the 1980s (Nakatsuka et al. 1981; Tomlinson et al. 1985; Hamaide and Emplit 1988; Rothenberg and Grischkowsky 1989; Rothenberg 1989), right after optical fiber solitons started to be actively investigated (Agrawal 2013). Indeed pulses in the form of solitons rely on the perfect balance between the effect due to the Kerr-induced self-phase modulation (SPM) and the anomalous (negative) group-velocity dispersion (GVD). However, in the opposite regime of normal (positive) GVD, the SPM enforces the dispersive broadening leading to steepening the pulse fronts. Under most common situations, such steepening was found to occur symmetrically on both the leading and trailing pulse fronts (Nakatsuka et al. 1981; Rothenberg and Grischkowsky 1989), at variance with regimes where it is only one of the two fronts that steepens due to mechanism of intensity-dependent velocity (Demartini et al. 1967; Grischkowsky et al. 1973; Anderson and Lisak 1983). The subsequent stage of steepening in fibers also features the onset of fast oscillations (Tomlinson et al. 1985; Hamaide and Emplit

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1988; Rothenberg and Grischkowsky 1989). At that time, however, the analogy with the breaking phenomena occurring in the spatial domain went unnoticed, nor it was properly realized that the wave-breaking regularized by GVD as observed in fibers was the manifestation of a general phenomenon predicted and investigated in other areas of physics. Indeed the study of DSWs was actually pioneered in the context of plasmas and fluids. In the 1960s indeed Sagdeev and coworkers pointed out the oscillatory nature of shock waves occurring in extremely rarefied plasmas (Moiseev and Sagdeev 1963) (this type of plasma is called collisionless, and hence at that time, DSWs have been mainly termed collisionless shocks). Few years later, in a celebrated seminal contribution, Zabusky and Kruskal (Zabusky and Kruskal 1965) numerically investigated (see also Trillo et al. 2016 for an experimental realization) the dispersive breaking of a sinusoidal wave in the framework of the Kortewegde Vries (KdV) equation in an attempt to give an explanation to the problem of recurrences in the Fermi-Pasta-Ulam problem (Fermi et al. 1965). In plasma the dispersive type of breaking was observed in ion acoustic waves few years later (Taylor et al. 1970). Dispersive breaking was also observed in water waves under the denomination of undular bores (Hammack and Segur 1974). On the theoretical side, the first construction of a DSW solution of a nonlinear dispersive model (again the KdV) was proposed by Gurevich and Pitaevskii (1974), who exploited the socalled Whitham averaging (1965) to predict the asymptotic evolution of a steplike initial condition (shock). Such an approach can be considerably extended to more general models (Dubrovin and Novikov 1983; Kamchatnov 2000). Among these, the defocusing nonlinear Schrödinger equation (NLSE) has received a considerable theoretical interest (Gurevich and Krylov 1987; Anderson et al. 1992; El et al. 1995; Kodama and Wabnitz 1995; Quiroga-Teixeiro et al. 1995; Forest and McLaughlin 1998; Kodama 1999), clearly establishing a link with the nonlinear propagation in optical fibers in the regime of normal GVD, which is described by such a model (Agrawal 2013). However it is only more recently that the understanding of the dispersive wave-breaking phenomena in fiber optics has been substantially advanced and placed in a more general context (Biondini and Kodama 2006; Finot et al. 2008; Fratalocchi et al. 2008; Trillo and Valiani 2010; Malaguti et al. 2010; Conti et al. 2010; Wabnitz 2013; Moro and Trillo 2014), at the same time achieving remarkable experimental results (Varlot et al. 2013; Fatome et al. 2014; Xu et al. 2016, 2017; Wetzel et al. 2016; Parriaux et al. 2017) and establishing a link with applications such as supercontinuum generation and frequency comb spectroscopy (Liu et al. 2012; Millot et al. 2016). Here, starting from the first principles, we will review the main theoretical and experimental results connected with dispersive breaking in fibers. These encompass the studies of DSWs in multiple four-wave mixing phenomena, the role of background in wave-breaking of pulses, the control of shock dynamics via simple Riemann waves, the photonic reproduction of the breaking of a dam, the competition of shock formation with other mechanisms of breaking, the prediction of radiation emitted by shocks, and the role of DSWs in nonconservative fiber environments (passive fiber resonators).

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An important point to underline is that shock waves in fiber optics literature are often associated with the steepening term that arises as a higher-order correction to SPM (Anderson and Lisak 1983; Agrawal 2013). This shock-driving term, however, becomes effective only for ultrashort pulses with sub-ps durations. Conversely Kerrinduced SPM is effective for longer time scales (ps or longer pulses or modulations in the range of tens of GHz). Indeed SPM, once acting alone, does not affect the temporal waveform of pulses, while it becomes responsible for strong steepening when acting in conjunction with weak GVD. The effect of initially weak GVD then becomes quite pronounced close to the steepened fronts, inducing dispersive wavebreaking characterized by smooth evolutions toward strongly oscillating envelopes. This mechanism is the leading-order effect that is at the basis of the phenomena discussed in this chapter. The chapter is organized as follows. In the following section, we review briefly the basic general concepts of classical shock waves and their regularization. Then, in the section entitled “Shock Formation in Optical Fibers,” we discuss the details of wave-breaking phenomena in fibers, with emphasis on recent advances. The photonic analogue of the process of the rupture of a dam is studied in the section entitled “Riemann Problem and Dam Breaking.” The section entitled “Competing Wave-Breaking Mechanisms” is devoted to show that certain regimes of wavebreaking can compete with modulational instability, while the problem of radiation is discussed in the section “Resonant Radiation Emitted by Dispersive Shocks.” Finally, the section “Shock Waves in Passive Cavities” briefly considers the case of passive fiber resonators and is followed by the general conclusions. Details on the mathematical construction of a DSW, according to the modulation theory, are reported in Appendix A.

Gradient Catastrophe and Classical Shock Waves The fundamental concepts related to shock waves can be introduced by considering the simplest extension to the nonlinear regime of the equation uz C cut D 0, i.e., the transport equation or linear unidirectional wave equation for the generic wave disturbance u D u.t; z/, where c is the linear velocity of the waves. Note that, as commonly used in fiber optics (as well as in other areas in the field of nonlinear waves Trillo et al. 2016), we assigned the role of independent variable to the distance z instead of time t , while we continue to refer to c as a velocity in the spirit of the theory of shock waves, though, in this notation, c has the dimension of an inverse velocity. By admitting that this velocity becomes proportional (or equal, for simplicity) to the local wave elevation u, i.e., c D c.u/ D u, we obtain the following equation known as the Hopf or inviscid Burgers equation:

uz C uut D 0

,

uz C ft .u/ D 0I f .u/ D

u2 ; 2

(1)

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which takes, as shown, the form of a scalar conservation law characterized by a particular form of the so-called flux f .u/, which in this case turns out to be quadratic in u. Here, for the time being, the Hopf equation (1) is introduced as a basic toy model, though we will go back to discuss it in the following, in order to show also its practical relevance for fiber optics. Equation (1) can be solved with the method of characteristics (Whitham 1974; LeVeque 2004), i.e., given a generic initial waveform u0 .t / D u.t; z D 0/, the generic value at time t0 , say uO 0 D u0 .t0 /, is transported along the linear characteristic t .z/ D t0 Cc.u0 .t0 //z D t0 C uO 0 z. Along a negative slope front, these characteristics are oblique convergent lines that determine a gradient catastrophe at the finite breaking distance zb where they first cross. For the Hopf equation, one easily finds zb D 1=.m/, where m is the maximal negative slope of u0 .t / (Whitham 1974). Beyond this distance the field becomes multivalued, as shown, as an example, in Fig. 1a for a Gaussian input. It is important to note that this mechanism is driven by the nonlinearity since, in the linear case such that c is a constant, all the characteristic lines are parallel and never cross each other (this is indeed the simple case in which any input wave is transported without deformation and moves with velocity c). A jump is introduced to overcome the problem of the multivalued u, as shown by the dashed vertical line in Fig. 1a. A classical shock wave is a piecewise smooth solution of the more general conservation law uz C ft D 0 which contains such a jump. The jump moves along a path ts .z/ according to the so-called RankineHugoniot (RH) condition, which gives the shock velocity: Œf  dts .z/ D ; dz Œu

Vs D

(2)

where Œ: : : is the contracted notation for the difference of the quantity inside parenthesis across the jump. Equation (2) is derived by considering that across the

a

0.5 zb

1

2

2

ts −2

ts(z)

z

u

z=0

0

b

0 t

2

1

0

−2

0 t

2

Fig. 1 (a) Gaussian input developing a gradient catastrophe at distance z D zb according to the Hopf equation. The dashed red line is the classical shock wave which has the (temporal) location ts D ts .z/. (b) Corresponding shock dynamics in the plane .t; z/. The first point where characteristic lines intersect stands for the gradient catastrophe point occurring at the breaking distance z D zb . The red dashed curve ts .z/ emanating from this point corresponds to the classical shock wave path

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a

b

1

2

Hopf u(t,z=6) KdV u(t,z=6) u(t,z=0)

1

u

u

Hopf u(t,z=6) KdV u(t,z=6) u(t,z=0)

0 −15

0

−1 −10

0

−5

5

10

−15

−10

0

−5

5

10

t

t

Fig. 2 Snapshots comparing the post-breaking evolutions ruled by the dispersionless (Hopf) and dispersive (KdV) model, respectively: (a) DSW formed by input u.t; 0/ D Œ1tanh.t /=2 (dashed), "2 D 0:05. (b) Input u.t; 0/ D sech.t / (dashed), "2 D 0:005 giving rise to a DSW and a rarefaction wave on the negative and positive slope front, respectively

jump the differential form (1) of the conservation law is undefined, whereas the associated integral form, Z t2 d u.t; z/dt D f .t1 ; z/  f .t2 ; z/; (3) d z t1 continues to hold for any t1 ; t2 and reduces to Eq. (2), assuming t1  ts  t2 (Whitham 1974). In general the RH condition (2) is equivalent to select one curve in the wedge of intersecting characteristics, as shown in Fig. 1b. On this basis, the initial jump from u.t  0/ D uL to u.t > 0/ D uR is a classical shock wave that moves, according to Eq. (2), with velocity Vs D .u2L =2  R u2R =2/=.uL  uR / D uL Cu . For instance, by specializing the previous result to a 2 unit amplitude step (uL D 1; uR D 0), the characteristic velocity turns out to be Vs D 1=2. We briefly recall that the additional condition uL > uR , usually known as entropy condition (originating from the language of gas dynamics), must hold for a shock solution to be valid (Whitham 1974). In the opposite case uL < uR , the solution that turns out to be compatible with the conservation law is a rarefaction wave. In the latter case, the step is smoothed out as it propagates (see Fig. 2b below for a visual example).

Regularization Mechanisms Classical shock waves constitute an important tool in many problems of fluid dynamics. However, in several applications the impact of dissipative or dispersive phenomena cannot be neglected. One can include dissipation in Eq. (1) by adding the lowest-order even derivative, which leads to the famous Burgers equation: uz C uut D ˛ut t ;

(4)

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which possesses, assuming boundary conditions u.t D 1/ D 1 and u.t D 1/ D 0, 1 the shock solution u D 12 f1 C tanhŒ 4˛ .t  12 z/g. This represents a smooth shock moving with velocity v D dt =d z D 1=2. Therefore the viscous effect introduces a finite width 4˛ of the shock, while the velocity remains equal to the velocity obtained in the inviscid case. Indeed, in the limit ˛ ! 0, such solution reduces to the unit jump (classical shock wave) solution of Eq. (1) with the same velocity vs D 1=2 predicted by Eq. (2). As a result, one can say that the classical shock wave is the zero dissipation limit of the shock wave propagating in the presence of losses. Vice versa dispersive regularization turns out to be more complex and somehow more intriguing. It can be understood by considering the lowest-order dispersive correction to Eq. (1), namely, the odd derivative ut t t weighted by the small coefficient "2 (the first odd derivative ut is trivially removable by introducing a shifted coordinate). In this case one obtains the famous integrable KdV equation (in the weak dispersion limit), which reads uz C uut C "2 ut t t D 0:

(5)

Even if "2  1, the dispersion drastically alters the dynamics. In Fig. 2, we compare the evolution ruled by the dispersionless (Eq. (1)) and the weakly dispersive (Eq. (5)) models, respectively. As shown in Fig. 2a, for a negative slope smooth front, the three-valued field ruled by Eq. (5) is regularized by dispersion through the onset of spontaneous oscillations, i.e., a nonstationary DSW that links the lower and upper constant states. In Fig. 2b we show the case of u.t; 0/ D sech.t /, which for the KdV stands for an initial value which contains no solitons. In this case, the DSW is accompanied by the formation of a rarefaction wave on the positive slope front. Importantly, (i) the period of the oscillations scales with dispersion being O."/, as evident by comparing Fig. 2a, b; this means that the limit " ! 0 the oscillation become infinitely dense, and hence one can recover the dispersionless limit only when averaging over the oscillating structure; (ii) the oscillations expand in a region bounded by the characteristic velocities of the leading and trailing edges of the DSW, which replace the single velocity Vs of the classical shock wave. These velocities can be obtained in the framework of Whitham modulation theory (Gurevich and Pitaevskii 1974; Whitham 1965; Hoefer et al. 2006; El and Hoefer 2016). The latter assumes the DSW to be constituted by a modulation of an invariant periodic solution of the underlying nonlinear dispersive model (the so-called cnoidal or d noidal wave, from the name of the corresponding Jacobian elliptic functions), characterized by slowly varying parameters (compared with the oscillation average period). Averaging over the oscillations gives the socalled Whitham equations which rule the evolution of such parameters. Self-similar smooth solutions of such equations allow to calculate the edge velocities and other parameters of interest for the DSW (details on the application of such theory to the NLSE are reported in Appendix A). This type of dispersive regularization of the shock and the general features discussed above are quite general for a wide class of models characterized by a small dispersion operator that perturbs the core dynamics ruled by a hyperbolic

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conservation law (or hydrodynamics model), which is responsible for driving the system toward the formation of a shock wave. In the proper regime of dispersion, the typical model that describes the nonlinear propagation in optical fibers also belongs to this class, as we discuss below. In this case the underlying conservation law turns out to be a vectorial one due to the fact that the wave envelope amplitude is complex. Before discussing the fiber case in detail, we emphasize that there are important physical situations such that the dissipative and the dispersive regularization mechanisms act jointly. Perhaps the simplest (yet retaining great physical importance) model that describes such situation is the so-called KdV-Burgers equation: uz C uut  ˛ut t C ˇut t t D 0;

(6)

which accounts for both the dissipative effect in Eq. (4) and the dispersive effect in Eq. (5) at once. Such model describes, for instance, ion acoustic waves in a dusty plasma. As demonstrated experimentally in Nakamura et al. (1999), in this case, the shock wave can have a prevalent oscillatory or smooth (monotonic) structure, depending on the relative weight of the coefficient ˛ and ˇ. As a matter of fact, one can calculate a precise threshold for the dissipation ˛, depending on the dispersion coefficient ˇ and the shock velocity, above which the regularized shock wave becomes monotonic (Karpman 1975). In fiber optics, an experimentally relevant situation which involves the interplay of dissipative and dispersive effects is that of an optical field recirculating in a passive cavity with synchronous reinjection and undergoing shock formation, a topic that will be briefly covered at the end of the chapter. Furthermore, we point out that, here (and in the remainder of this chapter), we have considered for sake of simplicity only models constituted by nonlinear and dispersive partial differential equations. However, shock waves and their regularized versions can develop also for discrete models, which describe the propagation of waves in lattices or oscillator chains. Among a large body of literature on this case, important results have been produced for the Hamiltonian models originally considered by Fermi-Pasta-Ulam (Fermi et al. 1965) or integrable versions such as the well-known Toda lattice. Dissipation can be additionally considered in such type of models, and shock waves have been explicitly discussed, for instance, in Hietarinta et al. (1995) for the Toda model and in Salerno et al. (2000) for the discrete NLSE (see also Cai et al. 1997 for a continuous version of such model), the latter being potentially of interest for coupled waveguide arrays, where coupling plays the role of effective dispersion in the system.

Shock Formation in Optical Fibers In general, a sufficiently accurate model for describing the propagation along an optical fiber of a pulse that modulates the a carrier at frequency !0 is the following envelope equation, namely, a generalized NLSE (a more general model can be

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employed to properly model the phenomenon of fiber supercontinuum (Agrawal 2013), but the following is sufficient for our goals): iEZ C i k 0 ET 

k 00 i .jEj2 E/T  TR .jEj2 /T E D 0; ET T C  jEj2 E C 2 !0

(7)

where the nonlinear terms are due, in order from left to right, to the SPM owing to the nearly instantaneous Kerr effect, the so-called self-steepening due to the dispersion of nonlinearity, and the intrapulse Raman scattering with coefficient TR , which is nothing but the characteristic time of the Raman response. It is important to emphasize that, in the literature, the self-steepening term is also termed the shock term, and indeed its coefficient is also indicated using the notation Tshock D 1=!0 , which we find misleading since DSWs can originate, as we show below, from SPM which should be regarded as the main shock-inducing term in the model, at least for common pulses with durations in the range of tens of picoseconds. Usually, the self-steepening term becomes important, in fiber optics, for much shorter (sub-psec) pulses. Conversely the effect of this term can be especially important in photonic crystal planar waveguides for longer pulses. As far as the dispersive terms are concerned, for the time being, we restrict ourselves to the effect of leading-order term or second-order dispersion or GVD with coefficient k 00 (higher-order effects will be further discussed with reference to radiation driven by shock waves). When the GVD is weak, the correct way to derive a hydrodynamic limit of Eq. (7) is to make use of the Wentzel-Kramers-Brillouin (WKB) method (or, equivalently, the Madelung or geometric optics transformation) applied to the equation cast in the following semiclassical form: i"

z

 ˇ2

"2 2

tt

C j j2

C i "S .j j2 /t  R .j j2 /t D 0;

(8)

where we have the normalized variables t D .T  k 0 Z/=T0 p conveniently introduced 00 and z D Z= Ld Lnl , ˇ2 D sign.k / D k 00 =jk 00 j, as well as the following smallness dispersion parameter and the normalized nonlinear coefficients: s "D

Lnl 1 Dp ; Ld N

s S D

P ; jk 00 j!0

R D

TR ; T0

(9)

where Lnl D . P /1 and Ld D T02 =jk 00 j are the nonlinear and dispersion length, respectively, N D Ld =Lnl is the soliton order, and P D max.jE.Z D 0; T /j2 / and T0 are the peak power and the duration of the input p envelope E.Z D 0; T /. By inserting in Eq. (8) the WKB ansatz u.t; z/ D .t; z/ expŒiS .t; z/=" and introducing the chirp u D St , we obtain   3 2 z C ˇ2 u C S  D 0; 2 t

(10)

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t t .t /2  uz C ˇ2 uut C Œ C S .u/  R t t D " 4  22 21

 :

(11)

t

Equations (10) and (11) without approximations are fully equivalent to the NLSE (8). The so-called dispersionless limit of the generalized NLSE (analogous of the Hopf equation for the KdV) is obtained by neglecting the RHS of Eq. (11), which is of higher-order [O."2 /] with respect to the LHS of the equations that 1 0 Incidentally, note that the term correspond to order O.  / and O. 2/, respectively.  1 in the RHS VQP D 4 t t =  .t / =22 is equivalent to the quantum potential in the interpretation of quantum mechanics based indeed on such potential (Bohm and Hiley 1984). In particular, in quantum mechanics, VQP acts along with the standard external potential, though being determined by the probability density , in the framework of the standard Schrödinger equation. In this “dispersionless” approximation, Eqs. (10) and (11) have the form of the evolution equations of a Eulerian fluid, with  and u playing the role of the fluid density (or water elevation) and velocity, respectively. Potentially, all the nonlinear terms can contribute to forming shock singularities. In particular, the steepening term is clearly a shockdriving term (see below and Agrawal 2013; Anderson and Lisak 1983), while the Raman term was shown to support smooth shock waves (or kink solitons) similarly to the KdV-Burgers model discussed above, both in the anomalous (Agrawal and Headley III 1992) and normal (Kivshar and Turitsyn 1993; Kivshar and Malomed 1993) dispersion regimes. Furthermore the effect of Raman response on the wavebreaking dynamics of ultrashort pulses has been also addressed in Quiroga-Teixeiro et al. (1995); Conti et al. (2010). However, one must realize that the coefficients of steepening and Raman terms are normally such that S ; R  1, i.e., they are much smaller than the Kerr coefficient (which is scaled to one in the adopted units), unless the regime of ultrashort pulses is considered (i.e., sub-psec down to few fsec). Therefore we will mainly address the DSWs developing through the Kerr effect which is of leading-order under usual experimental conditions, whereas we leave the concomitant effects of steepening and Raman as an advanced topic, which needs additional consideration and investigation, and will not be discussed here to a further extent. Nonetheless, the general form of Eqs. (10) and (11) is extremely useful to understand the impact of GVD over the formation of shock waves. In particular, if we consider the formal limit of strictly vanishing GVD (ˇ2 D 0), they reduces to the following system: z C 3S t D 0I

uz C ΠC S .u/  R t t D 0;

(12)

which shows that the power  turns out to be decoupled from the chirp (phase) dynamics, with the power evolution being ruled by a shock-bearing equation of the Hopf type. In the strict limit of zero GVD, therefore, one can conclude that the shock formation in the intensity profile is solely driven by the steepening term, thus occurring along the negative slope front of the pulse, as in the generic example of

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Fig. 1a (Anderson and Lisak 1983; Agrawal 2013). This is consistent with the fact that, in this limit, the Kerr SPM, which is responsible for the “pressure”-like term t in the second of Eqs. (12), only induces a phase modulation or chirp, thereby not affecting the temporal profile of the intensity. However, such conclusion becomes incorrect in the presence of an arbitrarily small dispersion, i.e., a GVD of order "2 (with " arbitrarily small) compared with the Kerr effect of order O.1/, as in Eq. (8). In fact, even if the RHS of Eq. (11) is dropped as in the dispersionless limit of the NLSE (8), the GVD affects the dynamics through terms that still appear in Eqs. (10) and (11), since they retain the same leading order of the nonlinear term. These terms turn out to be extremely important because they couple the SPM-induced chirp to the power . Therefore, when GVD is arbitrarily small but nonvanishing, the steepening of the pulse fronts can occur through the Kerr effect or SPM, which becomes the dominant one whenever S ; R  1. In this regime, the formation of the shock can be described in the framework of the reduced quasi-linear system of two equations (Gurevich and Shvartsburg 1970; Gurevich and Krylov 1987), commonly (though improperly) denoted also as the dispersionless NLSE:

z C ˇ2 .u/t D 0I

uz C ˇ2 uut C t D 0:

(13)

Shock formation occurs when Eqs. (13) are hyperbolic, i.e., in the regime of normal GVD, where ˇ2 D 1. In this case, the tendency to overtake is caused by the formation of a non-monotonic chirp which drives also the steepening of the power profile (Anderson et al. 1992), until the gradients become so large that the dispersive effects associated with the RHS in Eq. (11) set in. In the normal GVD regime (ˇ2 D 1), Eqs. (13) are identical to the shallow water equations (SWEs), which rule in 1D the propagation of the water elevation  and the (vertically averaged) horizontal velocity u, with interchanged role of space and time (in hydrodynamics the evolution variable is considered to be the time t , while the longitudinal distance plays the role of the time variable in fiber optics). In hydraulics these equations are also known as Saint-Venant equations. Interestingly enough, they also have one-to-one correspondence to the so-called p-system which rule the gas dynamics for an isentropic gas with pressure law p D 2 =2. Importantly, the SWEs can also be cast in the diagonal form in terms of new variables r ˙ .t; z/ D p u.t; z/ ˙ 2 .t; z/, which, in the language of the hyperbolic equation theory, are called Riemann invariants: rz˙ C V ˙ rt˙ D 0;

(14)

p where V ˙  V ˙ .r ˙ / D .3r ˙ C r  /=4 D u ˙  are the real eigenvelocities of the problem. An additional formulation of Eqs. (13) refers to the form of a differential conservation law, which allows for introducing the classical shock waves with their characteristic-associated velocities by means of extending the RH condition (Eq. (2)) based on the integral formulation of the conservation law. In this case the

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conservation law, which can be easily derived from Eqs. (13), is no longer a scalar one but rather the following 2  2 vectorial form: qz C Œf.q/t D 0;

(15)

where q D .; u/T and the flux turns out to be f.q/ D .ˇ2 u; ˇ2 u2 C 2 =2/T . Physically, Eqs. (15) express the conservation of mass and momentum in differential form. When a classical shock (a step) is introduced, the vectorial equivalent of the RH condition (2) can be introduced. However, since this becomes a vectorial condition which involves the same shock velocity Vs in the two components, it fixes not only the shock velocity but also the admissible values of the jump (i.e., the left .L ; uL / and right .R ; uR / values are not arbitrary but satisfy a constraint, a fact that bears no similarity with the scalar case) (LeVeque 2004; Hoefer et al. 2006; Conforti et al. 2014; Xu et al. 2017). The classical shock wave introduced in this way is crucial to develop a description of its dispersive regularization, i.e., of the DSW commonly observed in optical fibers. Furthermore, it is the key ingredient for properly describing the general evolution of steplike initial conditions (see below). In general, the fact that there are two eigenvelocities or two families of characteristics in the dispersionless NLSE (Eqs. (13), (14), and (15)) expresses the fact that the NLSE describes indeed the bidirectional type of dispersive hydrodynamics. The latter is characterized by two families of wave trains that have different characteristic speeds (which, in the laboratory frame, are naturally referred to the group velocity of light, which is removed in the passage from Eq. (7) to (8)) (El and Hoefer 2016). This is at variance with the KdV equation and its dispersionless limit that describes unidirectional hydrodynamics, as discussed in the previous section. Although Eqs. (13) rule the formation of shock wave via the mechanism of gradient catastrophe, there are important differences with the similar phenomenon ruled by a scalar conservation law (such as the Hopf equation). First, symmetric pulse envelopes used in several fiber optics applications typically exhibit two points of breaking instead of a single one. Only in some specific case these points degenerate in a single one (see discussion in the following subsection). Second, although there is a method, namely, the so-called hodograph transform (Whitham 1974; Moro and Trillo 2014), which permits to invert the role of dependent and independent variables in Eqs. (13), thus reducing the model to a linear model (LeVeque 2004; Whitham 1974), in practice, solutions are indeed difficult to be written. In general, for a generic smooth input waveform, it is challenging to predict the finite distance after which undergoes breaking. Apart from special cases for which this becomes possible (Moro and Trillo 2014), one must resort to (i) numerical simulations, (ii) approximate estimates (Anderson et al. 1992), and (iii) bind the breaking distance between a lower and upper value employing a general criterium due to Lax (Forest and McLaughlin 1998; Lax 1973). Below, we will discuss in more details the mechanisms of wave-breaking under different excitations in the regime described by Eqs. (13), which is the one widely observed in nonlinear fiber optics experiments in the psec regime.

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Mechanisms of Wave-Breaking in the Normal GVD Regime The shallow water equations (13), or their equivalent diagonal (Eqs. (14)) or conservation law (Eqs. (15)) formulations, entail two different main breaking mechanisms depending on the input pulse shape. For input bright and symmetric bell-shaped pulses 0 D .t; z D 0/ with a generic finite background, steepening occurs symmetrically (in contrast with the steepening ruled by the Hopf equation) until two points of gradient catastrophe occur at finite distance on both the leading and trailing edges of the input pulse. The strong gradients are regularized by the quantum potential term (RHS in Eqs. (10) and (11), which, in order to describe the post-breaking dynamics, is equivalent to reconsider the full dimensionless NLSE i"

z



"2 2

tt

C j j2

D 0:

(16)

The dynamics ruled by Eq. (16) is smooth at any distance. The initial stage, ruled by Eqs. (13), is dominated by the nonlinearity which causes a steepening of the tails. The physical mechanism behind such steepening is the fact that the dominant Kerr nonlinearity induces a strong self-phase modulation. The acquired phase turns out to be proportional to the intensity j 0 .t /j2 , in turn implying an instantaneous frequency deviation or chirp ı!.t / D @t .t / D @t j 0 .t /j2 . Such a chirp is turned into an instantaneous change of velocity ıu.t / / ı!.t /, according to the fact that in normally dispersive media, the velocity decreases for increasing frequency. As a consequence the top parts of the pulse experience a larger absolute velocity toward the pulse tails (the velocity is negative on the leading edge of the pulse and positive on the trailing edge, respectively), which is at the origin of the front steepening. Such process proceeds as the self-phase modulation grows along the fiber. However the point of infinite gradient (breaking) and successive overtaking is never reached. Indeed, as the gradient along the steepened front grows sufficiently large, the weak GVD becomes effective, and spontaneous formation of oscillations starts to become visible around the strongly steepened fronts. This usually occurs slightly before the breaking distance defined in the dispersionless limit ruled by Eqs. (13). The expanding oscillations form two symmetric DSWs, as shown in Fig. 3a, b for a Gaussian input. They propagate and expand in opposite directions compared with the natural group velocity of the wave, which corresponds the line t D 0 in Fig. 3. A mathematical description of each of the two DSWs is possible on the basis of Whitham modulation theory, which can be successfully formulated for the defocusing NLSE (16). According to such an approach, the DSW is described as a slowly varying modulation of the invariant nonlinear traveling-wave periodic solution (so-called d n-oidal or cn-oidal wave). The slowly varying parameters of such wave obey modulation equations which are obtained by performing a proper averaging due to Whitham (1965) over the fast oscillation. For the defocusing NLSE, the modulation equations are a set of four equations, which turns out to be hyperbolic and diagonalizable. A simple and smooth (rarefaction) solution of

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Fig. 3 False color level plot (a, c) and snapshots of power  D j j2 (b, d) of DSWs ruled by NLSE (16), " D 0:05: (a–b) Gaussian input 2 1=2 0 D Œ0:1 C 0:9 exp.t / with 10% power pedestal; (c–d) input 0 D sech.t /, with no pedestal

such equations allows to characterize the modulation and hence the DSW. We report in Appendix A the details of such calculations for the interested reader. An important point is that a single DSW is always characterized by two edges, as also clear from Fig. 3a. One of the edge is such that the amplitude of the oscillations vanishes: this is the linear edge of the DSW (it corresponds to the trailing edge of the right-going DSW or the leading edge of the left-going DSW; see Fig. 3a). Over the opposite edge of the same DSW, which exhibits the deepest oscillation, the periodic solution locally reduces to a soliton (i.e., the modulus m of the Jacobian function locally tends to one): hence, this is called the soliton edge of the DSW (it corresponds to the leading edge of the right-going DSW or the trailing edge of the left-going DSW; see Fig. 3a). The modulation theory allows to calculate the two velocities of the linear and soliton edges, respectively. However, it is worth pointing out that the modulation theory is an asymptotic method which provides a good description only

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for large enough propagation distances and/or small enough dispersion parameter ". In any event the dynamics of DSWs developing from smooth initial data cannot have strict correspondence with the DSW construction based on the modulation equations, since the latter implies an initial condition for the modulation equation in the form of a step (see Appendix A). Conversely, a quantitative test of the outcome of the modulation theory becomes possible by injecting steplike pulses, a case that we treat separately in the following. It is important to emphasize that the pulse background has a strong influence on the formation of the DSW, particularly in terms of the contrast of the oscillations. This is clear from Fig. 3, where Fig. 3c, d contrasts the evolution relative to the input 0 D sech.t / with the case of the Gaussian with background in Fig. 3a, b (note that, for a more rapidly decaying pulse than the hyperbolic secant, such as the Gaussian considered in Fig. 3a, b, the oscillations would be barely visible in the limit of vanishing background). This is the reason for which early measurements in fibers (Rothenberg and Grischkowsky 1989) reported a much lower contrast compared with later spatial experiments performed with non-zero background (Wan et al. 2007). A detailed experimental study of the effects of the background has been recently performed in Xu et al. (2016). Figure 4a summarizes the contrast of the oscillations (defined conventionally for the first fringe, as indicated in Fig. 4b), which are measured as a function of the background (in percent of the peak power of the pulse), while explicit examples for background levels corresponding to 0:1; 1; 6% are shown in Fig. 4b–g, comparing experimental results and NLSE simulations. As shown, the contrast of the oscillation is rapidly enhanced by increasing the background, reaching a saturated contrast of 100% where the minimum power of the leading edge touches zero. At larger background levels, the contrast smoothly decreases since such null point shifts from the leading edge of the DSW, a phenomenon that can be accurately described in the framework of the so-called dam-breaking experiment concerning steplike initial data (see section “Riemann Problem and Dam Breaking”). A physical insight into the role of the background can be gained as follows. The oscillating wave train in the DSW can be understood as the interference phenomenon between original components with zero chirp (the background) and new frequencies generated via the nonlinearity and GVD, which coexist on the temporal regions which correspond to the steepened tails of the pulse. The resulting pattern is characterized by the difference frequency and a visibility which depends on the amplitude ratio between such frequency components, thus requiring the background to be sufficiently large to give rise to large contrasts. When a dark input of the type 0 D tanh.t / is considered, the breaking occurs in t D 0. In contrast with the previous breaking mechanism (where only one Riemann invariant breaks at each catastrophe point), the latter is a nongeneric mechanism which implies breaking of both Riemann invariants in the null intensity point t D 0 (this can be shown analytically for the similar case for which the phase profile is suppressed, i.e., 0 D j tanh.t /j Moro and Trillo 2014). The emerging DSW exhibits a single fan with a narrower central black (zero-velocity) soliton and symmetric gray pairs around it (see Fig. 5a, b). The intermediate stage displayed in Fig. 5a, b shows the typical features of a DSW. However, in this case, one can

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100

Contrast [%]

80 (b,e)

60 40 20 (c,f) 0 0

10 5 Relative background level [%]

(d,g) 1

b

SIM.

max

e

15

EXP.

Power (a.u)

min

0 1 0 1

6%

c

6%

f 1%

d

0 -150

1%

g 0.1% 0 Time

150 -150

0.1% 0 Time

150

Fig. 4 Effect of pulse background on shock formation. (a) Contrast of the DSW fringe .Pmax  Pmin /=.Pmax C Pmin /, defined relative to the DSW leading edge, as shown in (b), versus percentage of the background (relative to the input peak power). (b)–(g): examples of different calculated and measured intensity profiles for 0.1, 1, and 6% background level: (b)–(d) numerics from NLSE and (e)–(g) experimental results (From Xu et al. 2016)

Fig. 5 (a,b) Wave-breaking occurring from 0 D tanh.t /, " D 0:005, from numerical integration of the NLSE: (a) false color level plot; (b) snapshots at z D 0 (input), z D 0:78 (breaking), z D 1:5 (DSW). (c) Snapshots of DSW formation from a gray input 0 D w tanh.wt / C i v, w2 C v 2 D 1. Here v D 0:5, " D 0:05

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further exploit the integrability of the NLSE and apply the inverse scattering method. This allows to show that the initial condition 0 D tanh.t / contains 2Ns  1 solitons, where Ns  b1="c (more precisely, one can show that 0 D tanh.t / turns out to be a reflectionless potential of the scattering problem associated with the NLSE whenever 1=" is integer). These solitons turn out to be a central black solitons and symmetric gray pairs with opposite velocities, which asymptotically (i.e., at normalized distance z  1=") separate on the same background (Fratalocchi et al. 2008). In this case the DSW is in fact a multisoliton solution, where the solitons embedded in the initial condition start to emerge after the catastrophe, which development is ruled by the SWEs or dispersionless NLSE. It worth pointing out that for the phase suppressed input ( 0 D j tanh.t /j), which shows the same type of breaking mechanism, only symmetric pairs of gray solitons emerge beyond the catastrophe point, while the central black soliton no longer appears in the DSW fan. Furthermore, similarly to the hyperbolic tangent case, a gray input still generates 2="  1 (narrower) gray solitons. However, in this case, the field breaks in two distinct points (see snapshot at z D 1:5 in Fig. 5c), from which two asymmetric DSWs emerge. These wave trains or solitonic DSWs can be shown to be asymptotically composed of 1="  1 and 1=" gray solitons, respectively. As discussed above the process of breaking depends on the shape of the input pulse, as shown in Figs. 3, 4, and 5. In particular, one can also find pulse shapes that do not lead to breaking. For instance, parabolic pulses are wave-breakingfree and led to the concept of similaritons, pulses which can be amplified while retaining their parabolic shape. On the other hand, other situations where wavebreaking phenomena lead to interesting developments involve a continuous wave with constant power and chirp modulation (Kodama 1999; Biondini and Kodama 2006; Wabnitz 2013). For instance, this leads to the generation of flaticon pulses, recently demonstrated in Varlot et al. (2013). Finally it is worth emphasizing that, in the spectral (Fourier) domain, the wavebreaking process is characterized by a strong spectral broadening. Indeed the steepening process corresponds to the generation of high frequencies in the spectrum. Beyond the breaking distance, the spectrum does not substantially reshape, while the DSW spreads (see Fig. 15b below, for an example of this behavior). In this regime, roughly speaking, the spectrum extends up to the highest frequencies of the generally non-monochromatic oscillations (for a different estimate of the broadening, see Parriaux et al. (2017). Such a dramatic spectral broadening has been shown to be highly beneficial for several applications such as supercontinuum generation and comb spectroscopy (Finot et al. 2008; Liu et al. 2012; Millot et al. 2016).

Shock in Multiple Four-Wave Mixing While in the previous subsection we have considered wave-breaking generated from bright or dark pulses, it was recently shown that DSWs can be generated also from periodic waves, i.e., amplitude-modulated waves which give rise, via the Kerr

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effect, to the phenomenon of multiple four-wave mixing (mFWM). This indicates the generation of multiple sideband orders at !0 ˙ n˝=2, n odd integer, which is produced via the Kerr term from an input beat signal or dual-frequency input !0 ˙ ˝=2 (Thompson and Roy 1991). Similarly an amplitude-modulated carrier frequency !0 , whose spectrum contains frequencies !0 ; !0 ˙ ˝, generates mFWM products !0 ˙ n˝, n integer. When such processes occur in the strong nonlinear regime which involves the generation of several mFWM orders, the field undergoes breaking. The wave-breaking mechanism turns out to be similar to the one illustrated in Fig. 5 for dark pulses. In order to understand the regime of breaking in mFWM, let us consider, for instance, the NLSE Eq. (16) subject to the initial condition p 2 cos.t =2/. This corresponds to the celebrated numerical experiment 0 D performed by Zabusky and Kruskal for the KdV equation (Zabusky and Kruskal 1965), though performed instead for the NLSE in the present case. In this case, without loss of generality, the normalized frequency detuning is fixed to ˝T0 D  by choosing T0 D 1=2f D =˝. p The dynamics is ruled, in this case, by the single smallness parameter " D f 4k 00 =. P / (Trillo and Valiani 2010). When "  1 the cosine input exhibits multiple points of breaking at the nulls of the power profile j 0 j2 D 2 cos2 .t =2/, i.e., at t D .2k C 1/, k D 0; 1; 2; : : :, as shown in Fig. 5a, b. The field at the wave-breaking points, displayed in Fig. 6b, is strongly reminiscent of the one generated in the hyperbolic tangent case shown in Fig. 5a. A remarkable difference is that, in the periodic case, the DSW emerging from each breaking point collides with the adjacent ones forming multiphase structures (see Fig. 6a, c). The elastic nature of the collision as well as the relationship between the darkness and the velocity of the single filaments that compose the DSW suggests that they behave as dark solitons. While solitons cannot exists in the strict sense due to the periodic nature of the problem, numerical results

Fig. 6 (a, b, c) Breaking of an input cosine according to NLS Eq. (16): (a) level colorplot of power j j2 ; (b–c) snapshots of power  D j j2 at breaking distance z D 0:34 (b) and z D 0:95 p (c) compared with the input (dashed blue). (d) Breaking for an input modulated field 0 D C p 2.1  / cos.t /, D 0:8. Here " D 0:04

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show that the finite-band solutions that arise in the scattering problem shrink in the limit of small ", resembling true solitons of the infinite line problem in the limit " ! 0. The number such soliton-like excitations grow as " decreases. Noteworthy, when the modulation has no zeros (i.e., for an imbalanced dualfrequency or triple-frequency input), the temporal locations of the breaking become non-degenerate, and breaking occurs at two distinct instants around all the minima of the input modulation, as shown in Fig. 5d. In this case, two symmetric DSWs emerge from each double breaking point, still giving rise to multiphase regions. The phenomenon of DSW in mFWM was observed in a recent fiber experiment, by employing the Picasso platform at Laboratoire Interdisciplinaire Carnot de Bourgogne in Dijon (Fatome et al. 2014). The setup allows to reach the necessary power of the input modulation at 28 GHz, without resorting to pulses, which would hamper the visibility of the DSW of the periodic case. Furthermore, since one cannot easily measure the field along fibers which are several km long, in the experiment, the evolution of the DSW is reconstructed at finite physical propagation length L D 6 km, by increasing the input power of the modulated field. Changing the power P at fixed length L (and fiber parameters k 00 and  , as well as fixedp modulation Ld Lnl D frequency) amounts indeed to change the normalized length z D L= p L k 00  P =T0 . The results are shown in Fig. 7. In particular the left column figures show the measured temporal traces versus the input power for three different configurations, corresponding to (a) sinusoidal input (suppressed carrier, balanced dualfrequency input), (b) dual-frequency (suppressed carrier) input with imbalanced power fractions, and (c) modulated carrier corresponding to triple-frequency input. In particular Fig. 7a confirms the scenario illustrated in Fig. 6a–c. In the other two cases, two breaking points arise around the minima of the modulated input, with preserved symmetry (in time) in the triple-frequency case and broken symmetry for the imbalanced dual-frequency (carrier suppressed) case. The simulations based on the dimensional NLS equation show a remarkable agreement in all cases (see Fig. 7d–f), without using any fitting parameter. Noteworthy, the NLSE does not need any higher-order corrections (steepening, Raman, higher-order dispersion) to correctly describe the experiment.

The Focusing Singularity We have extensively discussed the normal GVD regime case. However it is worth mentioning that also the anomalous GVD regime (k 00 < 0), which is described by a NLSE of the focusing type, can exhibit singularity formation. In this case the WKB reduction with ˇ2 D 1 takes the following form, obtained with the transformation u ! u, (Gurevich and Shvartsburg 1970): z C .u/t D 0I

uz C uut  t D 0;

(17)

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Fig. 7 DSWs arising from mFWM, showing the color level plot of the temporal profile of the modulated field against the input power P : experiment (a, b, c) vs. simulations (d, e, f) based on NLSE (7), neglecting steepening and Raman terms. (a–d) Sinusoidal (suppressed carrier balanced dual-frequency) input; (b–e) imbalanced dual-frequency (suppressed carrier) input; (c, f) modulated carrier (triple-frequency) input. The power is given in log units, P .dBm/ D 10 log10 P .mW / (From Fatome et al. 2014)

which shows the presence of the equivalent negative pressure term t . As a consequence, this dispersionless limit turns out to be elliptic nature, which reflects the fact that the full NLSE exhibits modulational instability (MI). In this case the catastrophe implies breaking around a focus point and is termed elliptic umbilic (see Dubrovin et al. 2015 and references therein). An example, which corresponds to the implicit solution of Eqs. (17) for the initial datum 0 D sech.t / discussed in Gurevich and Shvartsburg (1970), Kamchatnov (2000), is illustrated in Fig. 8a–c. In this case, the SPM induces a chirp which is initially linear around the origin and then increases its slope until eventually determines an abrupt change of sign around t D 0 (see Fig. 8a), somehow in a way similar to the null point for the 0 D tanh.t / initial datum in the defocusing regime of the NLSE (Conti et al. 2009). This, in turn, represents a compressional wave which induces a sharp peak with steepened

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a

c

z=0 z=0.35 z=0.5

0.2

u

0 −0.2

b ρ

4

2

0 −2

0 t

2

d

Fig. 8 Focusing catastrophe: (a, b) snapshots of chirp (hydrodynamical velocity) u (a) and power  (b) obtained from the dispersionless equations (Eq. (17)) with initial data 0 D sech2 .t /, u0 D 0, corresponding to NLSE with initial datum 0 D sech.t /;p (c) corresponding NLSE dynamics, " D 0:05; (d) similar dynamics for initial datum 0 D 2cos.t =2/ that leads to mFWM, " D 0:05

fronts at a finite critical distance (see cusp-like structure in Fig. 8b). It is worth pointing out that this dynamics is caused by the strongly dominant nonlinearity in the NLSE, thus being of different nature from the collapse phenomenon in the critical NLSE (i.e., the NLSE with cubic nonlinearity in 1 C 2 dimensions), which occurs right above the threshold (critical norm or power in the case of beams) where nonlinearity and diffraction mutually balance (soliton or so-called Townes profile). In the present case, dealing with 1 C 1D NLSE, the dynamics at the catastrophe and beyond it is regularized by the onset of dispersion and turns out to be usually very complicated (see Fig. 8c for an example) and can exhibit zones of multiphase oscillations separated by caustics ruled by a large ensemble of bright solitons (cases

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not shown) (Kamvissis et al. 2003). Only under particular conditions, which we will not deepen here, the observed oscillating structure represents the focusing analogue of the one-phase DSWs featured by the defocusing case. Incidentally, also input periodic waves, such as the cosine considered above for mFWM in the normal GVD regime (defocusing NLSE), exhibit the same type of breaking and similar post-breaking dynamics, except for the fact that now periodic (in time) points of breaking occur, as displayed in Fig. 8d. For long enough distances, adjacent structures are expected to collide and form even more complex patterns, which need further theoretical and experimental characterization. In the presence of noise, due to both high gain and large bandwidth which characterize the MI in the semiclassical (i.e., strongly nonlinear or weakly dispersive) limit, the impact of MI-amplified fluctuations could dramatically affect the breaking process (Ghofraniha et al. 2007). In this regime the breaking dynamics would strongly benefit from MI-suppressing mechanisms such as a nonlocal type of nonlinear response. The latter, however, is not effective in fibers, at least for long pulses (in the order of few tens or hundreds of psec; in fact, the retarded response due to the Raman scattering effect is the temporal analogue of a spatial nonlocality but becomes effective for shorter pulse durations). Overall, wave-breaking in the anomalous dispersion regime essentially remains an open area of investigation especially from the experimental point of view.

Control of DSW and Hopf Dynamics We finish this section by illustrating a regime of fiber optics propagation where the shock wave formation, though still being induced via the leading-order (Kerr) effect and in principle described by means of the NLSE, turns out to be effectively governed by the simpler Hopf or inviscid Burgers equation (1), introduced in the beginning of this chapter (Malaguti et al. 2010; Wabnitz 2013; Wetzel et al. 2016). If we consider pulses where the chirp (equivalent hydrodynamical velocity u) is p not arbitrary, but rather linked to the power  D j j2 as u D ˙2 , one of the two Riemann invariants in Eqs. (14) identically vanishes. Under this constraint, the solution of Eqs. (14) is called a simple Riemann wave (more generally a simple wave solution of a hyperbolic systems is such that one of the Riemann invariants is constant in the region of interest). In particular, in this case, it is easy to show that either the SWEs for power and chirp ; u or their diagonal form (14) for the Riemann invariant reduce to an effective equation of the Hopf type. In particular, for instance, the effective equation for the power  or the amplitude reads as (Malaguti et al. 2010) p z ˙ 3 t D 0

,

az ˙ aat D 0;

p a D 3 ;

(18)

i.e., a canonical Hopf equation for the normalized envelope amplitude a D a.t; z/. It is worth noting that, in this case, the Hopf equation holds for the pulse envelope

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amplitude (modulating the carrier !0 ). This is in contrast, for instance, with the dispersionless limit of the KdV equations introduced in the beginning of the chapter, where the variable is not an envelope variable but rather the wave amplitude itself in shallow water waves. Moreover, at variance with Eq. (12) where the steepening term dominates the dynamics, in Eq. (18), the Kerr effect prevails, and the validity of the model extends to regimes of relatively long pulses (i.e., psec to nsec). Noteworthy, the choice of the sign in Eq. (18), in turn fixed by the chirp sign, means that breaking can occur either on the negative (upper sign in Eq. (18)) or the positive (lower sign in Eq. (18)) slope front of the input waveform. According to Eq. (18), the dynamics p a pre-chirped pulse with phase initially R t of locked to .t; z D 0/ D ˙2"1 1 .t 0 ; z D 0/dt 0 is ruled by the implicit p p p solutions  D 0 .t  3 z/, u D ˙2 0 .t  3 z/ up to the point where the catastrophe occurs and dispersive effects appear. In this case the temporal symmetry implicit in the cases shown in Figs. 3 and 4 breaks down, since steepening occurs over one of the two fronts controlled by the choice of the upper or lower sign in the above solutions. Figure 9 displays the dynamics of an initially (positively) chirped bright pulse, contrasting the Hopf and the full NLSE dynamics. The pulse develops a gradient catastrophe over the trailing edge (negative slope front) at a distance of 500 m, where the characteristic lines of the Hopf equation shown in Fig. 9a first intersect. This dynamics has been recently confirmed in a fiber optics experiment (Wetzel et al. 2016). A further proposed generalization of this concept refers to additional tailoring of these Riemann waves by introducing a quadratic spectral phase by means of pulse shaping techniques. This allowed to experimentally prove the effectiveness of the method in order to tailor the distance of breaking, thus opening a promising route to shock wave control in optical fibers (Wetzel et al. 2016). At the bottom line of such result, it stands the noteworthy fact that a complicated model such as the nonlinear Maxwell equations which governs the propagation of the optical field is effectively reduced to a remarkably simple model such as the Hopf equation, which in the end rules the evolution of the electric field envelope. The validity of the Hopf model for appropriately pre-chirped waveform is obviously not limited to bright pulses. Figure 10a, b shows the case of the breaking of a dark profile (input power profile 0 D tan h2 .t /) with negative chirp, which still breaks on the trailing front, which, however, in this case represents the positive slope front. Such phenomenon is not restricted to any specific form of the pulse. Moreover, by combining different signs of chirp on the negative and positive temporal semiaxis, one can also enhance the rapidity of breaking compared with the unchirped case (case not shown, see Malaguti et al. 2010) or suppress the shock formation (see Fig. 10c). Therefore, not only in this regime, the nonlinear SPM induces a pure Hopf dynamics, but also it gives extra degrees of freedom to control the occurrence of breaking. Furthermore, in the presence of a tapering of the GVD (or equivalently third-order dispersion and frequency shifting), extreme highintensity compressed pulse can develop, which has been termed optical tsunamis in analogy to a shoaling tsunami in water waves (Wabnitz 2013).

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a

b

Intensity profiles

c

Chirp profiles

Fig. 9 (a) Intensity plot of the pulse profile of a Riemann simple wave propagating along a fiber obtained numerically from the NLSE model (top). The projected pulse intensity (bottom) is compared with the characteristic lines obtained analytically from the Hopf equation (white), showing shock formation at z D 500 m, where the characteristics start to intersect. Temporal profiles of the (b) intensity and (c) chirp are shown at selected distances, comparing predictions from the Hopf equation (solid black curve) with NLSE simulations (dashed blue curve) (From Wetzel et al. 2016, reporting the experimental realization of the Hopf dynamics. Here fiber parameters are k 00 D 0:8397 ps2 /km,  D 11:7 (W km)1 , and Gaussian pulse with 2:7 ps duration). (From Wetzel et al. 2016)

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Fig. 10 (a, b) Evolution and snapshot of a chirped input with hyperbolic tangent amplitude profile, undergoing a gradient catastrophe on the trailing edge at the normalized distance z  0:45 and the subsequent development into a DSW (note that a rarefaction wave develops on the leading edge); (c) shock suppression by a suitable choice of input chirp

Riemann Problem and Dam Breaking In the theory of quasi-linear hyperbolic PDEs such as the SWEs, a fundamental problem is the evolution of a steplike initial condition, which is known as the Riemann problem (LeVeque 2004). The solution to such problem is indeed a building block for understanding the scenario of possible evolutions as well as for developing numerical schemes of integration (Riemann solvers). According to the general theory, a steplike initial condition of a 22 problem, such as the SWEs, decays into a pair of fundamental waves, which can be of the shock wave or rarefaction wave type, respectively. In general a step initial condition involves a jump in both the variables  and u, which vary from the left .L ; uL / to the right .R ; uR / constant state, where, without loss of generality, the step is assumed to be located in t D 0, so that subscripts L and R refer to t < 0 and t > 0, respectively. In terms of Riemann invariants, one has also step initial conditions p p from the left values rL˙ D uL ˙ L to the right values rR˙ D uR ˙ R . According ˙ , the SWEs give rise to different to the specific value of the four boundaries rL;R evolutions which involve the decay into wave pairs of the type: (i) rarefactionrarefaction, (ii) shock-shock, and (iii) rarefaction-shock (LeVeque 2004; El et al. 1995). In particular, the latter case can be accessed by implementing only a step in the power variable  with the initial chirp being identically vanishing, which can be more easily accessed experimentally. In this case, the Riemann problem for the SWEs is known, in the context of hydrodynamics, as the dam-break problem, namely, the 1D evolution that follows the instantaneous removal (rupture) of a dam separating different downstream (L ) and upstream (R > L ) levels of still water. In this case the solution to the SWEs is composed by a rarefaction wave and a classical shock wave pointing in opposite directions (upstream and downstream, respectively) separated by a constant expanding plateau. While we refer the reader to the hydrodynamic literature for its derivation, Fig. 11 illustrates the profile of 

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a

Power [W]

1

Distance [km]

R

RW

classical SW 0.5

0

b

P

Z=15km

plateau

DSW PL −800

−600

−400

−200

0

200

400

15 DSW

plateau

RW τ4

τ3

10

τ2

τ1 τRH

5

(classical SW) 0

−800

−600

−400

−200 Time [ps]

0

200

400

Fig. 11 (a) Snapshots comparing in real units the DSW-RW pair (solid blue curve) obtained from the NLSE with steplike input E.T; 0/ D ŒPL C .PR  PL /.1 C tanh.T =Tr //=21=2 (dashed red) with the ideal dispersionless solution of the SWEs (solid black). The weak oscillations over the top of the RW should not be confused with those of the DSW, as they appear in the numerics for a fast jump (Tr small) due to the Gibbs phenomenon. (b) Wedges in real-world time-distance plane .T; Z/, corresponding to the RW (fable orange), p the DSW (green), and the plateau in between (white). The boundaries correspond to slopes k 00 PR j , j D 1; 2; 3; 4. Here k 00 D 176 ps2 /km,  D 3 (W km)1 , PR D 1 W, PL D 150 mW, and Tr D 10 ps, as in the experiment in Xu et al. (2017)

for such solution (see black solid line in Fig. 11a), contrasting it with the dispersive counterpart obtained from the numerical solution of the full NLSE (blue solid line in Fig. 11a) with steplike initial power profile (dashed red line in Fig. 11a). As shown, the solution arising from the SWEs or dispersionless limit gives a quantitatively good description of the NLSE dynamics for what concerns the smooth part that includes the rarefaction wave and the plateau. Conversely, the classical shock is replaced by a DSW as expected on the basis of the general principles illustrated in the introduction. In this situation, the edge velocities of the DSW can be predicted by applying the Whitham averaging theory (or modulation theory) illustrated in detail in the Appendix A. Following the general theory (see Appendix A), such velocities are conveniently expressed in terms of the self-similar variable D t =z. In the present case, the boundary conditions for the shock are given by the intermediate p p p p constant values of the plateau i D . L C R /2 =4, ui D L  R and the quiescent left state L ; uL D 0. This allows for expressing the linear (say

1 ) and the soliton (say 2 ) edge velocities as a function of L and R only, obtaining

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p

2 D 

L C 2

p

R

I 1 D

L  2R : p R

(19)

Conversely, the Whitham averaging gives for the edge velocities of the rarefaction wave, say 3 and 4 , the same expression which would be obtained in the dispersionless limit from the SWEs Eq. (13). They read as

4 D

p p 3 L  R p : R I 3 D 2

(20)

All of such velocities define, in the plane .t; z/, the wedges where the three components (rarefaction, plateau, and DSW) expand, as displayed in Fig. 11b. Importantly, unlike experiments performed with smooth pulse waveform, the dam-break problem gives the unique opportunity to quantitatively test the Whitham modulation theory against experimental results. In this respect, it is important to emphasize that, as illustrated in the Appendix A, the modulation theory predicts for the DSW a critical transition where the DSW envelope is no longer monotone (as it is in Fig. 11a) but rather exhibits a cavitating state associated with the appearance of a vacuum point where the optical intensity vanishes (see also Appendix A, Fig. 23). Such a transition occurs when the crucial parameter r D L =R , namely, the ratio between the two quiescent states, decreases below the threshold value r D rth D 1=9 ' 0:11. In the limit r ! 0 (L ! 0), the vacuum point shifts toward the linear edge of the DSW, but at the same time, the amplitude of the DSW oscillations vanishes. In this limit the shock disappears, and the dynamics involves a single rarefaction wave, recovering the solution of the SWEs in the so-called dry-bed case (a vanishing level of water downstream) (Kodama and Wabnitz 1995). The dam-breaking dynamics has been recently observed by means of a full fiber setup at the University of Lille (Xu et al. 2017). The input shape, obtained by means of an electrooptic generator driven by a generator of arbitrary waveform, is shown in Fig. 12a. This shape allows for observing the formation of the rarefactionDSW pair for the increasing step between the two states with nonvanishing powers PL and PR (leading edge of input pulse) while observing, at the same time, the dry-bed dynamics on the decreasing step from PR to zero (trailing edge of input pulse). The parameter of the fiber employed in the experiment is reported 2 in the caption of Fig. 11. The relatively large GVD (k 00 D 176 pps /km) permits 00 to have the oscillation period of the DSW, which scales as k = Pi , in the tens of psec range, allowing for a good temporal resolution of the fast oscillating DSW. Another crucial feature of the experiment is the compensation of the fiber losses, which would cause a strong deviation from the expected dynamics, by means of a counterpropagating Raman pump. Figure 12b shows the output profile after the propagation through the 15 km fiber. The measured profile clearly shows the rarefaction-DSW structure, which turns out to in good agreement with the NLSE simulation and with the delays calculated from modulation theory (dashed

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Fig. 12 Temporal traces of the whole waveform, experiment (blue dots) vs. numerics based on the NLSE (black curve; the input is in dashed red): (a) input power profile; (b) output power profile after propagation along a 15 km long fiber. The vertical dashed lines give the predicted delays of the edges of the DSW (magenta and green lines, from Eqs. (19)) and the RW (gray and orange lines, from Eqs. (20)), respectively

vertical lines). Conversely, only a rarefaction is observed on the trailing edge of the pulse. Figure 13 shows a detail of the DSW soliton edge, obtained for a fixed PR D 1 W and variable PL (variable ratio r). The measured profiles clearly show the critical transition to cavitation at the threshold value r D 0:11 (in full quantitative agreement with modulation theory), where the soliton at the edge of the DSW becomes black. Further decreasing r makes the vacuum point shifting toward the linear edge of the DSW, again in good quantitative agreement with modulation theory and NLSE simulations.

Competing Wave-Breaking Mechanisms Dispersive nonlinear wave propagation gives rise to different universal mechanisms of breaking. In addition to the formation of DSWs discussed so far, another wellknown breaking mechanism of a carrier wave due to growth of low-frequency modulations is the universal modulational instability phenomenon. For the scalar NLSE, these two mechanisms are mutually exclusive. In fact, the gradient catastrophe occurs in the defocusing regime characterized indeed by a hyperbolic dispersionless limit. Conversely, MI takes place in the focusing regime where the gradient catastrophe is precluded, reflecting the elliptic dispersionless limit of Eqs. (17) discussed above. However, there are several different situations where the two mechanisms can indeed coexist and compete. For example, they can coexist in the presence of higher-order dispersion (Conforti et al. 2014) or when nonlinearly coupled modes are considered as, for example, in the case of second harmonic generation for quadratic nonlinearities, or polarization modes for the Kerr nonlinearities.

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Fig. 13 (a–d) Zoom around the soliton edge of the DSW, showing the transition to cavitation, for fixed PR = 1 W and different fractions r D PL =PR : (a) 0.15; (b) 0.11 (cavitation threshold); (c) 0.07; (d) 0.03. Experiment vs. theory and vertical lines as in Fig. 12b

In nonlinear fiber optics, the vectorial model of interest that permits to observe the interplay between MI and DSW formation is the vector NLSE (VNLSE), which, adopting the same normalization that leads to the NLSE (8) can be cast in the following dimensionless form: i ".

j /z



"2 . 2

j /t t

 C j

jj

2

C Xj

3j j

2



j

D 0;

(21)

where j D 1; 2, and we assume to operate in the normal GVD regime. In contrast with the scalar case p (X D 0) which is modulationally stable, the plane-wave solutions uj D Pj expŒi .Pj C XP3j /z=" of Eqs. (21) are modulationally unstable provided X > 1. The MI gain reads as q p g D "1 K 2 Œ .P1  P2 /2 C 4X 2 P1 P2  .P1 C P2 /  K 2 I K 2  ."q/2 =2: (22) On the other hand, Eqs. (21) admit a hyperbolic dispersionless limit, as it is immediately clear in the symmetric case 1 D 2 for which Eqs. (21) reduces to a

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Fig. 14 Color level plot of j 1 j2 (j 2 j2 is similar) evolving according to Eqs. (21), comparing stable DSW formation for X D 1 (a), with the MI p unstable case for X D 2 (b). Here " D 0:04 and 1 .0; t / D 2 .0; t / D Œ0:1 C 0:9 exp.t 2 /= 2

scalar NLSE with nonlinear coefficient .1 C X /. Therefore a competition between DSW and MI should be expected in this case. Restricting for definiteness to the symmetric excitation of Gaussians on pedestal, Fig. 14 shows that this is indeed the case. Here, a crossover behavior is induced by changing the cross-phase modulation coefficient X at fixed " D 0:04. While for X D 1 (so-called Manakov case), the system exhibits the formation of stable DSW (Fig. 14a) in both modes, when X > 1 the onset of MI (Fig. 14b) leads to additional oscillations appearing on the flat top of the beams which dramatically affect the coherence of the DSW at large propagation distances.

Resonant Radiation Emitted by Dispersive Shocks Bright solitons propagating in standard or photonic crystal fibers close to the zerodispersion wavelength (ZDW) are known to emit resonant radiation (RR) in the region of normal GVD. The underlying mechanism is the resonant coupling with linear dispersive waves (DW) induced by higher-order dispersion (Akhmediev and Karlsson 1995). The emission of RR is usually thought to be a prerogative of solitons, but quite recently experimental observations (Webb et al. 2013; Conforti et al. 2015) and theoretical investigations (Conforti and Trillo 2013; Conforti et al. 2014) proved that this is not necessarily the case. In particular, the DSWs, which develop in the regime of weak dispersion, resonantly amplify DW at frequencies given by a specific phase-matching selection rule. In fact, the strong spectral broadening that accompanies wave-breaking seeds linear waves, which may be resonantly amplified, thanks to the well-defined velocity of the shock front. Consider, for example, a standard telecom fiber (Corning MetroCor) with nonlinear and dispersion parameters as follows:  D 2:5 W1 km1 , k2 D 6:4 ps2 /Km,

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Fig. 15 Temporal (a) and spectral (b) evolution of an input sech pulse P0 D 600 W, T0 D 850 fs, at p D 1568:5 nm (normal GVD). A/N labels anomalous/normal GVD regions, and the dashed red lines stand for the DW detuning predicted by Eq. (26) with velocity given by oblique dashed line in (a)

k3 D 0:134 ps3 /Km, and k4 D 9  104 ps4 /Km (higher-order terms are negligible), which gives a ZDW ZDW D 1625 nm (Webb et al. 2013). Figure 15 shows the temporal and spectral propagation of a hyperbolic secant pulse in the normal dispersion regime of the fiber, obtained, for the sake of completeness, from the numerical solution of generalized NLSE (gNLSE), which accounts for higherorder nonlinear terms in full integral form (Agrawal 2013). The pulse undergoes a steepening of the leading and trailing edge, which leads to wave-breaking at a propagation distance of 20 m. After the breaking, two DSWs develop with broken symmetry (in time) due to the presence of third-order dispersion. The spectrum is broadest at the breaking point and clearly shows a narrowband peak in the anomalous dispersion region. This peak can be interpreted as a resonant radiation emitted by the leading edge of the pulse. In the next section, we will show how to predict the spectral position of this resonant radiation.

Phase-Matching Condition We develop our analysis starting from the NLSE suitably extended to account for the effects of higher-order dispersion (HOD). In particular we extend the semiclassical form of NLSE (Eq. (8)) to include HOD terms, while we safely neglect Raman and self-steepening due to the pulse duration and power p range that we consider. By defining the dispersion coefficients as ˇn D @n! k= .Lnl /n2 .@2! k/n , we recover the defocusing NLSE in the weakly dispersing form (with ˇ2 D 1) i "@z

C d .i "@t /

C j j2

D 0; X ˇn .i "@t /n d .i "@t / D nŠ n2 D

"2 2 ˇ3 "3 3 ˇ4 "4 4 @t  i @ C @ C ::: 2 6 t 24 t

(23)

9 Shock Waves

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where we considered normal GVD. Note that the normalized dispersive operator d .i "@t / has progressively smaller terms, weighted by powers of the parameter "  1 and coefficients ˇn . The process of wave-breaking ruled by Eq. (23) can be described by applying p again the Madelung transformation D  exp .iS ="/. At leading-order in ", we obtain a quasi-linear hydrodynamic reduction, with  D j j2 and u D St equivalent density and velocity of the flow, which can be further cast in the form (Conforti et al. 2014)   ˇ3 2 ˇ4 3 z C ˇ2 u C u C u C : : : D 0; 2 6 t   ˇ ˇ 1 3 4 u3 C u4 C : : : C 2 D 0: .u/z C ˇ2 u2 C 2 6 2 t

(24) (25)

of a conservation law qz C Œf.q/t D 0 for mass and momentum, with q D .; u/, which suitably extends Eqs. (15). For small ˇ3;4 , since Eqs. (24) and 25) continue to be hyperbolic, thus admitting weak solutions in the form of classical shock waves, i.e., traveling jumps from left (L ; uL ) to right (R ; uR ) values, whose velocity Vc can be found from the generalized RH condition (i.e., the natural extension of Eq. (2) discussed previously in the scalar case: Vc .qL  qR / D Œf.qL /  f.qR / (LeVeque 2004). However, the jump is regularized by GVD in the form of a DSW. In this regime, the shock velocity can be identified with the velocity Vs of the steep front near the deepest oscillation (DSW leading edge), which differs from Vc and can be determined numerically (or analytically via modulation theory for steplike initial data, see Appendix A). The strong spectral broadening that accompanies steep front formation can act as an efficient seed for DW which are phase-matched to the shock in its moving frame at velocity Vs . In order to calculate the frequency of the DW, we assume an input pump .t; z D 0/ with central frequency !p D 0 (i.e., in real-world units !p 0 D coincides with !0 , around which d .i "@t / in Eq. (23) is expanded). Let us denote as Vs D dt =d z the “velocity” of the SW near a wave-breaking point and as dQ ."!/ D P ˇn n n nŠ ."!/ the Fourier transform of d .i "@t /. Linear waves exp.i k.!/z  i !t / are resonantly amplified when their wavenumber in the shock moving frame, which reads as k.!/ D 1" ŒdQ .!/  Vs ."!/ equals the pump wavenumber kp D k.!p D 0/ D 0. Denoting also as knl the difference between the nonlinear contributions to the pump and RR wavenumber (the nonlinear contribution to the wavenumber of the resonant radiation is induced by cross-phase modulation with a non-zero background, on top of which RR propagates), respectively, the radiation is resonantly amplified at frequency detuning ! D !RR that solves the explicit phasematching equation (Conforti et al. 2014) X ˇn n

nŠ

."!/n  Vs ."!/ D "knl :

(26)

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We show below that Eq. (26) correctly describes the RR emitted by a DSW. At variance with solitons of the focusing NLSE where Vs .!p D 0/ D 0 (Akhmediev and Karlsson 1995), DSWs possess non-zero velocity Vs , which must be carefully evaluated, having great impact on the determination of !RR .

Steplike Pulses We consider first a step initial value that allows us to calculate analytically the velocity. Without loss of generality, we take ˇ3 < 0. Specifically, we consider the evolution of an initial jump from the left state L ; uL D 0 for t < 0 to the right state p p R .< L /; ur D 2. R  L / for t > 0 (Hoefer et al. 2006; Conforti et al. 2014). The leading edge of the resulting DSW can be approximated by a gray soliton, p p p whose velocity can be calculated as Vl D L C uR D 2 R  L (Hoefer et al. 2006; Conforti et al. 2014). sol RR sol If we account for knl D knl knl D  1" L arising from the soliton knl D l =" RR and the cross-induced contribution knl D 2L =" to the RR, Eq. (26) explicitly reads as ˇ3 ˇ2 ."!/3 C ."!/2  Vs ."!/ C L D 0: 6 2

(27)

Real solutions ! D !RR of Eq. (27) correctly predicts the RR as long as jˇ3 j < 0:5, as shown by the NLSE simulation in Fig. 16. The DSW displayed in Fig. 16a clearly exhibits a spectral RR peak besides spectral shoulders due to the oscillating front, as shown by the spectral evolution in Fig. 16b. Perfect agreement is found between the RR peak obtained in the numerics and the prediction (dashed vertical line in Fig. 16b) from Eq. (27) with velocity Vs D VL , where VL is the leading or soliton edge velocity of the DSW that can be calculated by means of Whitham theory (see Appendix A). We also point out that knl represents a small correction, so !RR can be safelyq approximated by dropping the last term in Eq. (27) to yield

"!RR D "!RR D

3 .ˇ2 ˙ 2ˇ3 3ˇ2  ˇ3 (Webb

ˇ22 C 8Vs ˇ3 =3/, that can be reduced to the simple formula

et al. 2013) only when ˇ3 Vs ! 0.

Bright Pulses The behavior of steplike initial data are basically recovered for pulse waveforms that are more manageable in experiments. As shown in Fig. 17, RR occurs also in the limit of vanishing background, allowing us to conclude that a bright pulse does not need to be a soliton (as in the focusing NLSE, ˇ2 D 1) to radiate. In fact, resonant amplification of linear waves occurs via SWs also in the opposite regime where the nonlinearity strongly enforces the effect of leading-order dispersion, the

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Fig. 16 Radiating DSW ruled by NLSE (23) with " D 0:03, steplike input L ; R D 1; 0:5, and 3-HOD ˇ3 D 0:35: (a) Color level plot of density .t; z/ (the dashed line gives the DSW leading edge velocity Vl ); (b) corresponding spectral evolution. The vertical dashed line stands for the RR predicted by Eq. (27)

Fig. 17 (a, b) Temporal and spectral evolution of a Gaussian pulse without background emitting RR, for ˇ3 D 0:35, and " D 0:03

only key ingredients being a well-defined velocity of the front and the spectral broadening that seeds the RR at phase-matching. Experimental evidence for such RR scenario was reported quite recently (Webb et al. 2013), corresponding to numerical simulations reported in Fig. 15. The physical parameters used to obtain Fig. 15, gives normalized parameters " ' 0:07 and ˇ3 ' 0:37, typical of the wave-breaking regime ("  1) with perturbative 3-HOD. Since ˇ3 > 0, the radiating shock turns out to be the one on the leading edge (t < 0), and its velocity Vs D 0:75, inserted in Eq. (26), gives a negative frequency detuning fRR D !RR T01 =2 ' 13 THz, in excellent agreement with the value reported in Webb et al. (2013). A detailed numerical study of this particular case, including Raman effects is reported in Conforti and Trillo (2013).

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Periodic Input We are interested in the evolution ruled by Eq. (23) subject to the dual-frequency initial condition 0

D

p

exp.i !p t =2/ C

p 1  exp.i !p t =2/;

(28)

where we fix the normalized frequency !p  ˝T0 D  consistently with the notation introduced in the discussion of DSWs in mFWM. Here accounts for the possible imbalance of the input spectral lines. In this case a frequency comb is generated thanks to mFWM. Formation of DSWs, occurring in the regime of weak normal dispersion (ˇ2 D 1), enhances the broadening of the comb toward high-order sideband pairs. When such DSWs are excited sufficiently close to a ZDW, they are expected to generate RR, owing to phase-matching with linear waves induced by higher-order dispersion. An example of the RR ruled by TOD (we set ˇ3 D 0:3) is shown in Fig. 18 for an imbalanced input ( D 0:3 in Eq. (28)). The colormap evolution in Fig. 18a clearly shows that the initial waveform undergoes wave-breaking around z  0:4. The mechanism of breaking has been discussed before in section “Shock Formation in Optical Fibers,” and also analyzed in details in Fatome et al. (2014), Trillo and Valiani (2010). It involves two gradient catastrophes occurring across each minimum of the injected modulation envelope. The GVD regularizes the catastrophes leading to the formation of two DSWs, where the individual oscillations in the trains exhibits dark soliton features, moving with nearly constant darkness and velocity inversely proportional to it. Importantly, the breaking scenario is weakly affected by TOD; However, one can notice that the darkest soliton-like oscillation emits RR. This radiation has much higher frequency than the comb spacing and turns out to be generated over the CW plateau of the leading edge labeled 0 . This is clear from Fig. 18b, which shows the enhancement

Fig. 18 RR emitted by shock with asymmetric pumping D 0:3, and " D 0:04: (a) temporal and (b) spectral colormap evolution for ˇ3 D 0:3; Dashed line in (a) highlights the DSW edge velocity V D 0:4. Vertical dashed line in (b) indicates !RR from Eq. (26) with knl D j 0 j2 ="

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of such frequency at the distance of breaking where the strong spectral broadening associated with the shock acts as a seed for the phase-matched (resonant) frequency.

Shock Waves in Passive Cavities Passive nonlinear cavities which are externally driven have been recently become popular, and have been implemented both in fiber rings and monolithic microresonators. They are exploited for fundamental studies as well as impactful applications such as the formation of wide-span frequency combs. Such resonators exhibit extremely rich dynamics characterized by a host of phenomena such as bistability, modulational instability and soliton formation. It has been recently shown that passive cavities admits a novel dynamical behavior featuring the formation of dispersive-dissipative shock waves (Malaguti et al. 2014). These phenomena are well captured by a mean field approach which yields a driven damped NLSE, often referred to as the Lugiato-Lefever equation (LLE) Lugiato and Lefever (1987). By accounting for HOD, a generalized LLE can expressed in dimensionless units as i"

z

C d .i "@t /

C j j2

D Œı  i ˛

C iS;

(29)

where we adopt the normalization We just p introduced in Malaguti et al. (2014). L=Ld  1 (L and Ld D T02 =k 00 are the recall that the parameter " D fiber (cavity) length and the dispersion length associated with time scale T0 and GVD) quantifies the P smallness of the GVD and the HOD introduced through the operator d .i "@t / D n2 ˇn .i "@t /n =nŠ D ˇ2 "2 @2t =2  i ˇ3 "3 @3t =6 C : : :, where p the coefficients ˇn D @n! k= .L/n2 .@2! k/n [note that ˇ2 D sign.@2! k/] are related to real-world HOD @n! k. Let us first neglect the HOD terms (ˇn D 0, n > 2). Equation (29) can exhibit a bistable response with two coexisting stable branches of CW solutions. A DSW can be seen as a fast oscillating modulated wave train that connects two sufficiently different quasi-stationary states. Starting from a cavity biased on the lower state, one can easily reach a different state on the upper branch by using an addressing external pulse with moderate power. In this regime the intracavity pulse edges undergo initial steepening, which is mainly driven by the Kerr effect, tending to form shock waves. The strong gradient associated with the steepened fronts enhances the impact of GVD, which ends up inducing the formation of wave trains that connect the two states ofpthe front. p An example is reported in Fig. 19, by using the injected field S .t / D P C Pp sech.t / with P D 0:0041 and Pp D 0:16P . As shown for the conservative case in the previous Section, the presence of HOD may lead the shock to radiate. We concentrate on the first relevant dispersive perturbation, i.e., third-order dispersion (TOD ˇ3 ¤ 0), but the scenario is qualitatively similar for others order of HOD. An example is shown in Fig. 20, for the same injected field used for the example reported Fig. 19. In the temporal domain, the effect of TOD is to induce an asymmetry between the leading and

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Fig. 19 DSW generation ruled by Eq. (29). (a) Color level plot of intracavity power j .z; t /j2 . (b) Snapshots of intracavity power at difference distances. Here " D 0:1, ı D =10, ˛ D 0:03

Fig. 20 Temporal (a) and spectral (b) evolution of a DSW in the cavity. Parameters: ˇ2 D 1, " D 0:1, ı D =6, ˇ3 D 0:25, and ˛ D 0:03. Dashed blue line in (a) represents the front velocity Vs D 1; dashed line in (b) stands for the value of "!RR D 9:6 calculated from the phase-matching equation (30)

trailing fronts. The most striking feature is visible in the spectral propagation shown Fig. 19b, where an additional frequency component, well detached from the shock spectrum, is generated starting at a distance z 2. The frequency of this radiation can be found by means of a perturbation approach, in close analogy to the one developed for the conservative case. In the limit of small losses ˛, we find that the frequency of the RR must satisfy the following phase-matching equation:   ."!/3 ."!/2 ."!/ C ˇ2  ˇ3  ı C 2PuH D 0; 6 2 Vs

(30)

where PuH is the power of the higher state of the front, propagating with velocity Vs where RR is shed. This equation is very similar to Eq. (27), but it contains the cavity detuning ı as an additional parameter.

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Conclusions In summary, we have shown that nonlinear fiber optics represents an ideal playground for observing dispersive hydrodynamics phenomena and specifically for making accurate experiments on dispersive shock waves. Recent theoretical studies have permitted to substantially progress the full understanding of the phenomenon. Moreover, challenging experiments have been carried out that have demonstrated the accessibility of wave-breaking phenomena in different new contexts, ranging from multiple four-wave mixing to pulse shaping experiment, supercontinuum and comb generation, observation of radiative effects, and fundamental problems of fluid flow such as dam breaking. Yet, a lot of new experiments can be envisaged that could aim at unveiling new fundamental scenarios discussed in this chapter as well as to exploit wave-breaking in fiber optics applications. Furthermore optical experiments are important also for the understanding similar phenomena of hydrodynamic origin in other contexts ranging from tidal bores in fluid dynamics, gravity waves in the atmosphere, spin waves in magnetic films, and atom collective behavior in BoseEinstein condensates, for which the experimental implementation in the lab turns out to be more challenging.

Appendix A In this Appendix we outline the calculation of the power (density) and chirp (velocity) profile of the DSW, according to Whitham modulation equation and the construction originally proposed by Gurevich-Pitaevskii-Krylov (Gurevich and Pitaevskii 1974; Gurevich and Krylov 1987). Our approach closely follows that of Hoefer et al. (2006), to which the reader is also referred to for further details. Below, we specifically focus on the case of a right-going DSW, which is generated by a suitable choice of the initial step. We recall, however, that also a left-going DSW is a suitable solution of the NLSE, which can be easily obtained with symmetry arguments from the case discussed below. Let us start by considering an invariant traveling-wave periodic solution of the NLSE (Eq. (16)) of the form: .t; z/ D

p . / exp Œi . / ;

(31)

z where  tV is a fast variable since "  1. By means of direct substitution into " the NLSE, one can easily obtain the invariant dn-oidal solution:

.t; z/ D 3  .3  1 /dn2

p 3  1 jm ;

p 1  2 3 u.t; z/ D  D V  ;  0

(32) (33)

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whichpdepends on the four parameters 1 ; 2 ; 3 ; V , with the additional p constraint V D 1 C 2 C 3 , and D ˙1. Here the wave period L D 2K.m/= 3  1 1 is given in terms of the elliptic integral of first kind K.m/, with m D 23  . 1 A modulation of such dn-oidal solution describes the DSW. The slow (compared to L) evolution of the parameters of such modulation is ruled by the Whitham equations, obtained by averaging the conservation laws of the NLSE over the period L (Whitham 1965). For integrable systems such as the NLSE, these equations are known to be expressible in diagonal form. For the NLSE this can be done by introducing four Riemann invariants ri D ri .t; z/, i D 1; 2; 3; 4, r1 < r2 < r3 < r4 , which are a suitable combination of the original four parameters V; i (Pavlov 1987): @ri @ri C vi .r1 ; r2 ; r3 ; r4 / D 0I @z @t

i D 1; 2; 3; 4:

(34)

Here the velocities vi D vi .r1 ; r2 ; r3 ; r4 / constitute a deformation of the velocity V that depends on combinations of fri g and elliptic integrals of first (i.e., K.m/) and second (i.e., E.m/) kind. For instance, the velocity v3 D v3 .r1 ; r2 ; r3 ; r4 / that will be relevant in the following reads as   1 .r4  r2 /E.m/ 1 v3 D V  .r4  r3 / 1  ; 2 .r3  r2 /K.m/

(35)

and all quantities are recast as functions of ri , viz.: V D

1 .r1 C r2 C r3 C r4 /; 4

3 D

1 .r1  r2 C r3 C r4 /2 ; 16

2 D

1 .r1 C r2  r3 C r4 /2 ; 16

1 D

1 .r1  r2  r3 C r4 /2 ; 16

mD

.r4  r3 /.r2  r1 / : .r4  r2 /.r3  r1 /

(36)

We are interested to describe the DSW ruled by the NLSE with initial step data corresponding to

.t; z D 0/ D

L ; t < 0 ; R ; t > 0

u.t; z D 0/ D

uL ; t < 0 : uR ; t > 0

(37)

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A general step in amplitude decays into a combination of rarefaction and dispersive shock wave (El et al. 1995). In order to generate a pure right-going DSW, p we consider that the initial value in terms of Riemann invariants r ˙ D u ˙ 2  is characterized by r  D const:, i.e., it is a right-going simple wave. The condition of a decreasing step (L > R ) implies that the simple wave is of shock type (and not a rarefaction wave). In order to respect the simple wave assumption, we have to calculate one of the four parameters (L ; R ; uL ; uR ) as a functions of the others. p p By imposing, for example, uR D uL C 2. R  L /, the initial value in terms of Riemann invariants (see Fig. 21) is characterized by a constant value for r  and a decreasing steplike variation for r C : p r  .t; z D 0/ D uL  2 L ; ( p rLC D uL C 2 L ; t 0

(38) (39)

which lead to a traveling (right-going) SW that connects the two constant states (Hoefer et al. 2006; LeVeque 2004). The corresponding DSW can be described in terms of a self-similar simple rarefaction wave of Whitham equations (34) generated by the following four-dimensional initial value that arises from initial data regularization (Hoefer et al. 2006; Kodama 1999) (see Fig. 21): p r1 .t; z D 0/ D r  D uL  2 L I

Riemann invariants

r

4

r+L

r

3

L

r

s

+ R

r2 rr1 =t/z

Fig. 21 Sketch of the Riemann invariants of Eqs. (34), where r1 ; r3 ; r4 stay constant and r3 D r3 . D t =z/ varies smoothly in the range S   L between the values rRC and rLC . L and S correspond to the linear and soliton edges of the DSW, respectively. The dashed red lines are the initial Riemann invariants in the dispersionless limit

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p p r2 .t; z D 0/ D rRC D uL C 4 R  2 L I p r4 .t; z D 0/ D rLC D uL C 2 L I ( C p p rR D uL C 4 R  2 L ; t < 0 r3 .t; z D 0/ D p rLC D uL C 2 L ; t > 0:

(40)

In particular, the initial value (40) evolves in such a way that the Riemann variables r1 ; r2 ; r4 remain constant and only r3 D r3 . / varies, forming a pure rarefaction wave (owing to the fact that r3 .z D 0/ is non-decreasing) that depends on the selfsimilar variable D t =z. Indeed all Whitham equations are formally satisfied when r1;2;4 .t; z/ D const: and r3 .t; z/ D r3 . /, provided the equation .  v3 / r30 D 0 is fulfilled. For r3 . / ¤ const ant , this implies

D v3 .r  ; rRC ; r3 . /; rLC /:

(41)

Equation (41) is a nonlinear equation in the only unknown r3 . /, which can be solved with any root-finding method to compute the profile of the rarefaction wave. Once the value of r3 . / is known, the values of the parameters i and V are calculated from Eqs. (36), which can be plugged into Eqs. (32) and 33) to get the DSW profile. It can be easily proved that the power and the chirp of the DSW are bounded by the envelopes ˙ and u˙ defined as C .t; z/ D 2 ;  .t; z/ D 1 ; uC .t; z/ D V  u .t; z/ D V 

p p

1 3 =2 ; 2 3 =1 :

(42)

An example of DSW profile is shown in Fig. 22a–c and contrasted with numerical solution of NLSE in Fig. 22b–d, with a smoothed input step in order to avoid numerical artifacts. The velocities L and S of the leading (soliton) and trailing (linear) edges of the DSW correspond to the edges of this rarefaction wave and can be calculated as the limits of v3 .r  ; rRC ; r3 . /; rLC / for r3 ! rRC (m ! 1, soliton edge) and r3 ! rLC (m ! 0, linear edge), respectively:

L D lim V3 .r  ; rRC ; r3 ; rLC / D

.r   rLC /.rRC  rLC / 2rLC C r  C rRC C 4 2rLC  r   rRC

S D lim V3 .r  ; rRC ; r3 ; rLC / D

2rRC C r  C rLC 4

r3 !rLC

r3 !rRC

(43)

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415 b

1

1

0.8

0.8

0.6

0.6

ρ

ρ

a

0.4

0.4

0.2

0.2

0

0

0.5

1

1.5

0

2

0

0.5

t c

1.5

2

1.5

2

t d

0

0 -1

u

-1

u

1

-2

-2 -3

-3

-4

-4 0

0.5

1

t

1.5

2

0

0.5

1

t

Fig. 22 Power and chirp of a DSW calculated from Eqs. (32–33) (a, c) and corresponding NLSE simulation (b, d). Magenta dashed lines depict the envelopes Eqs. (42). Parameters: L D 1, R D 0:4,  D 0:03, z D 1. Vertical dashed lines indicate the position of the soliton and linear edge of the DSW obtained from Eqs. (43)

These values are reported as vertical dashed lines in Fig. 22. The modulation theory entails a crossover between two different regimes separated by a critical condition. In the first regime, the DSW envelopes are monotone and the DSW power never vanishes. However, below a critical value of L =R , the DSW exhibits a self-cavitating point, i.e., zero power, corresponding to a vacuum point in gas dynamics. Since the minimum power of the cnoidal wave turns out to be min D .r1  r2  r3 C r4 /2 =16 (Hoefer et al. 2006), the existence in the DSW of a cavitating or vacuum state min D 0 requires r3 D r1  r2 C r4 . Therefore we obtain the self-similar location of the vacuum by performing the limit r3 ! r   rRC C rLC in Eq. (35):

v D V3 .r  ; rRC ; r   rRC C rLC ; rLC / D " #1 r  C rLC rRC  r  rLC  rRC E.mv /  D 1 2 2 r   2rRC C rLC K.mv / !2 rRC  r  mv D rLC  rRC

(44)

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a

b

10

1 5

0.6

0

u

r

0.8

0.4 -5

0.2 0 -0.5

0

0.5

1

1.5

-10 -0.5

2

t

0

0.5

1

1.5

2

t

Fig. 23 Power (a) and chirp (b) of a DSW calculated from Eqs. (32) and (33) in presence of a vacuum point. Magenta dashed lines depict the envelopes Eqs. (42). Parameters: L D 1, R D 0:15,  D 0:03, z D 1. Vertical black dashed lines indicate the position of the soliton and linear edge of the DSW obtained from Eqs. (43), whereas the red dashed line highlights the position of the vacuum point from Eq. (44)

It can be shown that in order to get a solution that conserves momentum u, the velocity should change sign at the vacuum point (Hoefer et al. 2006). This fixes in Eq. (33) as:

.t; z/ D sign.t =z  v /: The threshold for cavitation can be found by imposing that the vacuum point, coincides with the soliton edge, or mv D 1, that gives 

R L

D th

1 : 4

(45)

At such threshold the vacuum point coincides with the soliton edge of the DSW, while decreasing the ratio R =L below the threshold makes the vacuum point progressively move along the envelope of the DSW toward its linear edge, where the oscillations vanish (this behavior has been experimentally confirmed for the Riemann dam-break problem, where the DSW develops between the lower constant state and the intermediate level generated in the dynamics; see section “Riemann Problem and Dam Breaking”). An example of non-monotonic DSW which features a vacuum point along its envelope is shown in Fig. 23a–b for R =L D 0:15.

References G.P. Agrawal, Nonlinear Fiber Optics, 5th edn. (Academic, New York, 2013) G.P. Agrawal, C. Headley III, Kink solitons and optical shocks in dispersive nonlinear media. Phys. Rev. A 46, 1573 (1992)

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A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

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Contents The List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solitons in Dual-Core Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Symmetry-Breaking Bifurcation (SBB) of Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . Gap Solitons in Asymmetric Dual-Core Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Coupler with Separated Nonlinearity and Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . Two Polarizations of Light in the Dual-Core Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solitons in Linearly Coupled Fiber Bragg Gratings (BGs) . . . . . . . . . . . . . . . . . . . . . . . . . . Bifurcation Loops for Solitons in Couplers with the Cubic-Quintic (CQ) Nonlinearity . . . Dissipative Solitons in Dual-Core Fiber Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Exact SP (Solitary-Pulse) Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special Cases of Stable SPs (Solitary Pulses) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability of the Solitary Pulses and Dynamical Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interactions Between Solitary Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CW (Continuous-Wave) States and Dark Solitons (“Holes”) . . . . . . . . . . . . . . . . . . . . . . . . Evolution of Solitary Pulses Beyond the Onset of Instability . . . . . . . . . . . . . . . . . . . . . . . . Soliton Stability in PT (Parity-Time)-Symmetric Nonlinear Dual-Core Fibers . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The chapter provides a survey of (chiefly, theoretical) results obtained for self-trapped modes (solitons) in various models of one-dimensional optical B. A. Malomed () Faculty of Engineering, Department of Physical Electronics, School of Electrical Engineering, Tel Aviv University, Tel Aviv, Israel ITMO University, St. Petersburg, Russia e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 G.-D. Peng (ed.), Handbook of Optical Fibers, https://doi.org/10.1007/978-981-10-7087-7_70

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waveguides based on a pair of parallel guiding cores, which combine the linear inter-core coupling with the intrinsic cubic (Kerr) nonlinearity, anomalous groupvelocity dispersion, and, possibly, intrinsic loss and gain in each core. The survey is focused on three main topics: spontaneous breaking of the inter-core symmetry and the formation of asymmetric temporal solitons in dual-core fibers; stabilization of dissipative temporal solitons (essentially, in the model of a fiber laser) by a lossy core parallel-coupled to the main one, which carries the linear gain; and stability conditions for PT (parity-time)-symmetric solitons in the dual-core nonlinear dispersive coupler with mutually balanced linear gain and loss applied to the two cores.

The List of Acronyms 1D: one-dimensional 2D: two-dimensional BG: Bragg grating CGLE: complex Ginzburg-Landau equation CQ: cubic-quintic (nonlinearity) CW: continuous-wave (solution) GPE: Gross-Pitaevskii equation GS: gap soliton GVD: group-velocity dispersion MI: modulational instability NLSE: nonlinear Schrödinger equation PT : parity-time (symmetry) SBB: symmetry-breaking bifurcation SP: solitary pulse SPM: self-phase modulation VA: variational approximation WDM: wavelength-division multiplexing XPM: cross-phase modulation

Introduction One of basic types of optical waveguides is represented by dual-core couplers, in which parallel guiding cores exchange the propagating electromagnetic fields via evanescent fields tunneling across the dielectric barrier separating the cores (Huang 1994). In most cases, the couplers are realized as twin-core optical fibers (Digonnet and Shaw 1982; Trillo et al. 1988) or, in a more sophisticated form, as twin-core structures embedded in photonic crystal fibers (Saitoh et al. 2003). Such double fibers can be drawn by means of an appropriately shaped preform from melt, or fabricated by pressing together two single-mode fibers, with the claddings removed

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in the contact area. Alternatively, a microstructured fiber with a dual guiding core can be fabricated and used too (MacPherson et al. 2003). If the intrinsic nonlinearity in the cores is strong enough, the power exchange between them is affected by the intensity of the guided signals (Jensen 1982). This effect may be used as a basis for the design of diverse all-optical switching devices (Friberg et al. 1987, 1988; Heatley et al. 1988; Królikowski and Kivshar 1996; Tsang et al. 2004; Uzunov et al. 1995; Lederer et al. 2008) and other applications, such as nonlinear amplifiers (Malomed et al. 1996a; Chu et al. 1997), stabilization of wavelength-division-multiplexed (WDM) transmission schemes (Nistazakis et al. 2002), logic gates (Wu 2004), and bistable transmission (Chevriaux et al. 2006). Nonlinear couplers also offer a setup for efficient compression of solitons by passing them into a fiber with a smaller value of the groupvelocity dispersion (GVD) coefficient: as demonstrated in work (Hatami-Hanza et al. 1997), the highest quality of the soliton’s compression is achieved when two fibers with different dispersion coefficients are not directly spliced one into the other, but are connected so as to form a coupler (a necessarily asymmetric one, in this case). In addition to the simplest dual-core system, realizations of nonlinear couplers have been proposed in many other settings, including the use of the bimodal structure (orthogonal polarizations) of guided light (Trillo and Wabnitz 1988), semiconductor waveguides (Villeneuve et al. 1992), plasmonic media (Hochberg et al. 2004; Petráˇcek 2013; Smirnova et al. 2013), and twin-core Bragg gratings (Mak et al. 1998; Tsofe and Malomed 2007; Sun et al. 2013), to mention just a few. In addition to the ubiquitous Kerr (local cubic) nonlinearity of the core material, the analysis has been developed for systems with nonlinearities of other types, including saturable (Peng et al. 1994), quadratic (alias second-harmonicgenerating) (Mak et al. 1997; Shapira et al. 2011), cubic-quintic (CQ) (Albuch and Malomed 2007), and nonlocal cubic interactions (Shi et al. 2012). Unlike the Kerr nonlinearity, more general types of the self-interaction of light can be realized not in fibers (i.e., not in the temporal domain), but rather in planar waveguides (i.e., in the spatial domain). Theoretical modeling of these settings is facilitated by the fact that the respective nonlinear Schrödinger equations (NLSEs) for the evolution of the local amplitude of the electromagnetic waves takes identical forms in the temporal and spatial domains, with the temporal variable replaced by the transverse spatial coordinate, in the latter case. Generally, dual-core systems are adequately modeled by systems of two linearly coupled NLSEs, in which the linear coupling represents the tunneling of electromagnetic fields between the cores (Jensen 1982; Wright et al. 1989; Snyder et al. 1991). Further, effective discretization of continuous nonlinear couplers may be provided, in the spatial domain too, by the consideration of parallel arrays of discrete waveguides (Herring et al. 2007; Hadžievski et al. 2010; Shi et al. 2013). The coupler concept was also extended for the spatiotemporal propagation of light in dual-core planar waveguides, with the one-dimensional (1D) NLSE replaced by its two-dimensional (2D) version, which includes both the temporal and spatial transverse coordinates (Dror and Malomed 2011).

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Similar to the double-fiber waveguides for optical waves are dual-core cigarshaped (strongly elongated) traps for matter waves in atomic Bose-Einstein condensates (BECs) (Strecker et al. 2003). Transmission of matter waves in these settings have been studied theoretically (Gubeskys and Malomed 2007; Matuszewski et al. 2007; Salasnich et al. 2010), making use of the fact that the Gross-Pitaevskii equation (GPE) for the mean-field wave function of the matter waves in BEC (Pethick and Smith 2008) is actually identical to the NLSE for electromagnetic waves in similar optical waveguides. The abovementioned systems in optics and BEC imply lossless propagation of optical and atomic waves; hence the respective models are based on the NLSEs and GPEs which do not include dissipative terms. On the other hand, loss and gain play an important role in many optical systems, such as fiber lasers. The fundamental model of these systems is based on complex Ginzburg-Landau equations (CGLEs), i.e., an extension of the NLSE with real coefficients replaced by their complex counterparts (van Hecke 2003; Grelu and Akhmediev 2012). Accordingly, dualcore fiber lasers are described by systems of linearly coupled CGLEs (Sigler and Malomed 2005). Dissipative linearly coupled systems with the gain and loss applied to different cores are relevant too, as models admitting stable transmission (Malomed and Winful 1996; Atai and Malomed 1996, 1998b), filtering (Chu et al. 1995a), and nonlinear amplification (Chu et al. 1997) of optical pulses in fiber lasers with the cubic nonlinearity; see a brief review of the topic in Malomed (2007). A special system is one with exactly equal gain and loss acting in the parallelcoupled cores, which are identical as concerns other coefficients (Driben and Malomed 2011a). Such settings feature the PT (parity-time) symmetry between the cores (for the general concept of the PT symmetry; see original works (Bender and Boettcher 1998; Ruschhaupt et al. 2005; El-Ganainy et al. 2007; Berry 2008; Musslimani et al. 2008; Makris et al. 2008; Klaiman et al. 2008; Longhi 2009) and reviews (Bender 2007; Makris et al. 2011; Suchkov et al. 2016; Konotop et al. 2016)). The linear spectrum of the PT -symmetric coupler may remain purely real (i.e., it does not produce decay due to imbalanced loss or blowup due to imbalanced gain), provided that the gain-loss coefficient does not exceed a critical value (in fact, it is exactly equal to the coefficient of the linear coupling between the cores (Driben and Malomed 2011a; Alexeeva et al. 2012); see section “Dissipative Solitons in Dual-Core Fiber Lasers” below). Overall, PT -symmetric settings may be considered as dissipative systems which are able to emulate conservative ones, as they support not only real spectra but also stable soliton families if appropriate nonlinearity is included (Musslimani et al. 2008; Suchkov et al. 2016; Konotop et al. 2016), as is shown below in detail in section “Dissipative Solitons in Dual-Core Fiber Lasers” (generic dissipative systems create isolated nonlinear states (in particular, dissipative solitons Malomed 1987b), which play the role of attractors, rather than continuous families of stable solutions). Thus, nonlinear dual-core systems represent a vast class of settings relevant to optics, BEC, and other areas, which offer a possibility to model and predict many physically significant effects. As examples of similar systems which are well known in completely different areas of physics, it is relevant to mention

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tunnel-coupled pairs of long Josephson junction, which are described by systems of linearly coupled sine-Gordon equations (see original works (Mineev et al. 1981; Kivshar and Malomed 1988; Ustinov et al. 1993) and reviews (Makhlin et al. 2001; Savel’ev et al. 2010)), and the propagation of internal waves in stratified liquids with two well-separated interfaces, which are described by pairs of linearly coupled Korteweg de Vries equations, which were derived in various forms (Gear and Grimshaw 1984; Malomed 1987a; Lou et al. 2006; El et al. 2006; EspinosaCeron et al. 2012). The purpose of this chapter is to present a reasonably compact review of the corresponding models and results. Because the general topic is very broad, the review is limited to optical waveguides based on dual-core optical fibers. Related settings, such as those based on double planar waveguides and double traps for matter waves in BEC, are briefly mentioned in passing. A fundamental property of nonlinear couplers with symmetric cores is the symmetry-breaking bifurcation (SBB), which destabilizes obvious symmetric modes (sometimes called supermodes, as they extend to both individual cores, which support individual modes), and gives rise to asymmetric states. The SBB was theoretically analyzed in detail for temporally uniform states (alias continuous waves, CWs) in dual-core nonlinear optical fibers (Snyder et al. 1991) and, in parallel, for self-trapped solitary waves, i.e., temporal solitons in the same system (Wright et al. 1989; Trillo et al. 1989; Paré and Florja´nczyk 1990; Maimistov 1991; Chu et al. 1993; Akhmediev and Ankiewicz 1993a; Soto-Crespo and Akhmediev 1993; Tasgal and Malomed 1999; Chiang 1995; Malomed et al. 1996b), as well as for dual-core nonlinear fibers with Bragg gratings (BGs) written on each core (Mak et al. 1998). Some results obtained in this direction were summarized in early review by Romagnoli et al. (1992) and later in Malomed (2002). The SBB analysis was then extended to solitons in couplers with the quadratic (Mak et al. 1997) and CQ (Albuch and Malomed 2007) nonlinearities. The Kerr nonlinearity in the dual-core system gives rise to the subcritical SBB for solitons, with originally unstable branches of emerging asymmetric modes going backward (in the direction of weaker nonlinearity) and then turning forward (for the classification of bifurcations, see Iooss and Joseph 1980). The asymmetric modes retrieve their stability at the turning points. On the other hand, the supercritical SBB gives rise to stable branches of asymmetric solitons going in the forward direction. For solitons, the SBB of the latter type occurs in twin-core Bragg gratings (Mak et al. 1998) (see section “Two Polarizations of Light in the Dual-Core Fiber” below) and in the system with the quadratic nonlinearity (Mak et al. 1997). The coupler with the intra-core CQ nonlinearity gives rise to a closed bifurcation loop, whose shape may be concave or convex (Albuch and Malomed 2007) (see details below in section “Solitons in Linearly Coupled Fiber Bragg Gratings (BGs)”). In models of nonlinear dual-core couplers, the SBB point can be found in an exact analytical form for the system with the cubic nonlinearity (Wright et al. 1989), and the emerging asymmetric modes were studied by means of the variational approximation (VA) (Paré and Florja´nczyk 1990; Uzunov et al. 1995; Peng et al. 1994; Mak et al. 1997, 1998; Dror and Malomed 2011) and numerical calculations

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(Akhmediev and Ankiewicz 1993a; Soto-Crespo and Akhmediev 1993); see also reviews Romagnoli et al. (1992) and Malomed (2002). In addition to the studies of solitons in uniform dual-core systems, the analysis was developed for fused couplers, in which the two cores are joined in a narrow segment (Sabini et al. 1989). In the simplest approximation, the corresponding dependence of the coupling strength on coordinate z may be represented by the delta function, ı.z/. Originally, interactions of solitons with a locally fused segment were studied in the temporal domain, viz., for bright (Sabini et al. 1989; Chu et al. 1995b; Mandal and Chowdhury 2005) and dark (Afanasjev et al. 1997) solitons in dualcore optical fibers and fiber lasers (Boskovic et al. 1995). In that case, the coupling affects the solitons only in the course of a short interval of their evolution. The spatial-domain optical counterpart of the fused coupler is provided by a dual-core planar waveguide with a narrow coupling segment created along the coordinate (x) perpendicular to the propagation direction (z) (Akhmediev and Ankiewicz 1993b; Li et al. 2012). Dynamics of spatial optical solitons in such settings, including stationary solitons trapped by the fused segment of the coupler and scattering of incident solitons on one or several segments, was analyzed in Harel and Malomed (2014). The objective of this chapter is to present a review of basic findings produced by studies of models developed for dual-core nonlinear optical fibers and fiber lasers, along the abovementioned directions. The review is chiefly focused on theoretical results, as experimental ones are still missing for solitons, in most cases. In section “Solitons in Dual-Core Fibers,” the most fundamental results are summarized for the SBBs of solitons in dual-core fibers with identical cores. Some essential findings for asymmetric waveguides, with different cores, are included too. The results are produced by a combination of numerical and methods and analytical approximations (primarily, the variational approximation, VA). In section “Dissipative Solitons in Dual-Core Fiber Lasers,” the results are presented for the creation of stable dissipative solitons in models of fiber lasers, the stabilization being provided by coupling the main core, which carries the linear gain, to a parallel lossy one. Section “Soliton Stability in PT (Parity-Time)-Symmetric Nonlinear Dual-Core Fibers” is focused on nonlinear dual-core couplers featuring the abovementioned parity-time (PT ) symmetry, which is provided by creating mutually balanced gain and loss in two otherwise identical cores, the main issue being stability conditions for the corresponding PT -symmetric solitons (which can be done in an exact analytical form, in this model). The chapter is concluded by section “Conclusion.”

Solitons in Dual-Core Fibers The Symmetry-Breaking Bifurcation (SBB) of Solitons The Formulation of the Model The basic model of the symmetric coupler, i.e., a dual-core fiber with equal dispersion and nonlinearity coefficients in the parallel-coupled cores, is represented

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

427

by a system of linearly coupled NLSEs, which are written here in the scaled form (Trillo et al. 1988), with subscripts standing for partial derivatives: 1 i uz C u  C juj2 u C Kv D 0; 2 1 i vz C v  C jvj2 v C Ku D 0; 2

(1) (2)

where z is the propagation distance,   t  z=Vgr is the reduced time (t is the physical time, Vgr is the group velocity of the carrier wave (Agrawal 2007)), and u and v are amplitudes of the electromagnetic waves in the two cores; sign C in front of the group-velocity-dispersion (GVD) terms, represented by the second derivatives, implies the anomalous character of the GVD in the fiber (Agrawal 2007), the cubic terms represent the intra-core Kerr effect, and K, which is defined to be positive (actually, it may be scaled to K  1), is the coupling constant accounting for the light exchange between the cores. An additional effect which can be included in the model represents the temporal dispersion of the inter-core coupling, represented by its own real coefficient, K 0 . The accordingly modified Eqs. (1) and (2) take the form (Chiang 1995) 1 i uz C u  C juj2 u C Kv C iK 0 v D 0; 2 1 i vz C v  C jvj2 v C iKu C K 0 u D 0: 2

(3) (4)

Below, this generalization of the nonlinear-coupler model is not considered in detail, as the analysis has demonstrated that the dispersion of the inter-core coupling does not produce drastic changes in properties of solitons (Rastogi et al. 2002). Equations (1) and (2) can be derived, by means of the standard variational procedure, from the respective Lagrangian (L), which, in turns, includes the Hamiltonian of the system (H ) (Malomed 2002): 

  i   u uz C v  vz d  C c:c:  H; 2 1  Z C1    1    1 juj4 C jvj4  K u v C uv  ; H D ju j2 C jv j2  2 2 1 Z

C1

LD

(5) (6)

where both  and c.c. stand for the complex-conjugate expressions. The Hamiltonian, along with the integral energy (alias total norm) of the solution, 1 ED 2 and the total momentum,

Z

C1

1

 2  juj C jvj2 d ;

(7)

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B. A. Malomed

P D

i 2

Z

C1

1



 uu C vv C c:c: d ;

(8)

are dynamical invariants (conserved quantities) of the system. If the dispersion of the inter-core dispersion is included, see Eqs. (3) and (4), the additional term in the Hamiltonian density in Eq. (6) is  .iK=2/ .u v C v  u   uv  vu , while the expression for the total energy and momentum keep the same form as defined in Eqs. (7) and (8). Equations (1) and (2) admit obvious symmetric and antisymmetric soliton solutions (supermodes), u D ˙v D a1 sech

  a

exp

 iz ˙ iKz ; 2a2

(9)

where a is an arbitrary width, which determines the total energy (7), and Hamiltonian (6) of the symmetric and antisymmetric states 2 Esymm-sol D 2a1 ; Hsymm-sol D  a3  4Ka1 : 3

(10)

In the case of K > 0 (that may always be fixed by definition), the antisymmetric solitons are unstable (Soto-Crespo and Akhmediev 1993), as they correspond to a maximum, rather than minimum, of the coupling term (K) in Hamiltonian (10), therefore they are not considered below. For the symmetric solitons, the SBB, which destabilizes them and replaces them by stable asymmetric solitons, with different energies in the two cores, is an issue of major interest. The onset of the SBB of the symmetric soliton, i.e., the value of the soliton’s energy at the SBB point, can be found in an exact analytical form (Wright et al. 1989). To this end, one looks for a general (possibly asymmetric) stationary solution of Eqs. (1) and (2) for solitons with propagation constant k as fu .z;  / ; v .z;  /g D e ikz fU ./; V ./g ;

(11)

where real functions U and V satisfy the following ordinary differential equations: 1 d 2U C U 3 C K V D kU; 2 d2

(12)

1 d 2V C V 3 C KU D kV: 2 d2

(13)

The SBB corresponds to the emergence of an antisymmetric eigenmode of infinitesimal perturbations, fıU . /; ıV . /g D " fU1 . /; U1 . /g

(14)

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

429

(" is a vanishingly small perturbation amplitude) around the unperturbed symmetric solution of Eqs. (12) and (13), which is taken as per Eq. (9), i.e., U0 . / D V0 . /  a1 sech

  a

; kD

1 C K: 2a2

(15)

The linearization of Eqs. (12) and (13) around the exact symmetric state leads to the equation 1 d 2 U1 C 3U02 U1  .K C k/ U1 D 0; 2 d2

(16)

which is tantamount to the stationary version of the solvable 1D linear Schrödinger equation with the Pöschl-Teller potential. Then, with the help of well-known results from quantum mechanics (Landau and Lifshitz 1989), it is easy to find that, with the growth of the soliton’s energy E, i.e., with the increase of k (see Eqs. (10) and (15)), a nontrivial eigenstate, produced by Eq. (16), appears at Wright et al. (1989) p p E D Ebif  4 K=3  2:31 K:

(17)

Thus, the SBB and destabilization of the symmetric solitons (15) take place precisely at point (17).

Continuous-Wave (CW) States and Their Modulational Instability (MI) To complete the formulation of the model of the symmetric nonlinear coupler, it is relevant to mention that, besides the solitons, it admits simple continuous-wave (CW) states, with constant U and V in Eq. (11). Indeed, Eqs. (12) and (13) easily produce the full set of CW solutions: symmetric and antisymmetric ones, (CW) (CW) D Vsymm D Usymm

p

.CW/

(CW) k  K; Uanti D Vanti

D

p

k C K;

(18)

which exist, respectively, at k > K and k > K. With the growth of k, i.e., increase of the CW amplitude, the symmetric state undergoes the SBB at k D 2K, giving rise to asymmetric CW states, which exist at k > 2K: s (CW) Uasymm

D

k C 2

r

k2 4

s  K 2;

(CW) Vasymm

D

k  2

r

k2  K2 4

(19)

(and its mirror image, with U  V ). The SBB for CW states in models of couplers with more general nonlinearities was studied in detail in Snyder et al. (1991). However, all the CW states are subject to the modulational instability (MI) (Trillo et al. 1989), which, roughly speaking, tends to split the CW into a chain of solitons. While this conclusion is not surprising in the case of the anomalous GVD in Eqs. (1) and (2), as it gives rise to the commonly known MI in the framework of the single

430

B. A. Malomed

NLSE (Agrawal 2007), all symmetric and asymmetric CW states are modulationally unstable too in the system of linearly coupled NLSEs with the normal sign of the GVD in each equation (Tasgal and Malomed 1999), this MI being produced by the linear coupling. Because of the instability of the CW background, nonlinear couplers, even with normal GVD, cannot support stable dark solitons or domain walls, i.e., delocalized states in the form of two semi-infinite asymmetric CWs, which are transformed into each other by substitution U  V , linked by a transient layer (Malomed 1994). Because bright solitons cannot exist in the case of the normal GVD, the development of the MI in the latter case leads to a state in the form of “optical turbulent” (Tasgal and Malomed 1999). As concerns the temporal dispersion of the inter-core coupling (see Eqs. (3) and (4)), its effect on the MI of the CW states was studied in Li et al. (2011).

The Variational Approximation (VA) for Solitons Asymmetric solitons, which emerge at the SBB point, cannot be found in an exact form, but they can be studied by means of the VA. This approach for solitons in nonlinear couplers was developed in works (Paré and Florja´nczyk 1990; Maimistov 1991; Chu et al. 1993); see also work (Ankiewicz et al. 1993) which discussed limitations of the VA in this setting. Here, the main findings produced by the VA and their comparison with results obtained by means of numerical methods are presented as per work (Malomed et al. 1996b). The VA is based on the following trial analytical form (ansatz) for the two-soliton soliton: u D A cos. /sech

 

 exp i . C

a    exp i .  v D A sin. /sech a

 / C i b 2 ;

(20)

 / C i b 2 ;

(21)

where real variational parameters, A,  , a, , , and b, may be functions of the propagation distance, z. In particular, A.z/ and a.z/ are common amplitude and width of the two components, chirp b.z/ must be introduced, as it is well known (Anderson 1983; Anderson et al. 1988), in the dynamical ansatz which allows evolution of the soliton’s width, .z/ is an overall phase of the twocomponent soliton, angle .z/ accounts for the distribution of the energy between the components, and .z/ is a relative phase between them. The shape and phase parameters form conjugate pairs, viz., .A; /, . ; /, and .a; b/. Note that the ansatz based on Eqs. (20) and (21) assumes that centers of the two components of the soliton are stuck together. This implies that the linear coupling between the two cores is strong, which corresponds to the real physical situation. Nevertheless, it is also possible to consider a case when the linear coupling plays the role of a small perturbation, making a two-component soliton a weakly bound state of two individual NLSE solitons belonging to the two cores (Abdullaev et al. 1989; Kivshar and Malomed 1989b; Cohen 1995).

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

431

Switching of a soliton between the two cores of the coupler was considered, on the basis of a full system of variational equations for ansatz (20), (21) in work (Uzunov et al. 1995). It was also demonstrated by Smyth and Worthy (1997) that the approximation for the switching dynamics in the nonlinear coupler can be further improved if the radiation component of the wave field is incorporated into the ansatz. Here, the consideration is focused on the basic case of static solitons, for which ansatz (20), (21) gives rise to the following variational equations, in which all parameters of ansatz (20) and (21), but overall phase , are assumed constant: sin.2/ sin.2 / D 0; E cos.2/  K cot.2/ cos.2 / D 0; 3a   1 a1 D E 1  sin2 .2/ ; 2 d 1 2E D  2C dz 6a 3a

(22) (23) (24)

 1 1  sin2 .2/ C  sin.2/ cos.2 /; 2

where E is the soliton’s energy, which, according to its definition (7), takes value E D A2 a for ansatz (20), (21). As it follows from Eq. (22), the static soliton may have either sin.2/ D 0 or sin.2 / D 0. According to the underlying ansatz, the former solution implies that all the energy resides in a single core, which contradicts Eqs. (1) and (2), hence this solution is spurious. The latter solution, sin.2 / D 0, implies that cos.2 / D ˙1. As mentioned above, the solutions corresponding to cos.2 / D 1, i.e., antisymmetric ones, with respect to the two components, are unstable. Therefore, only the case of cos.2 / D C1, corresponding to solitons with inphase components, is considered here. Then, width a can be eliminated by means of Eq. (24), and the remaining equation (24) for the energy-distribution angle  takes the form of  cos.2/

  E2 1 sin.2/ 1  sin2 .2/  1 D 0: 3K 2

(25)

Further analysis reveals that, in the interval 0 < E 2 < E12 , where p E12 D .9=4/ 6K  5: 511 K;

(26)

the only relevant solution to Eq. (25) is the symmetric one, with  D =4 (corresponding to cos .2/ D 0) and equal energies in both components, according to Eqs. (20) and (21). When the soliton’s energy attains value E1p , predicted by Eq. (26), there emerge asymmetric solutions with cos.2/ D ˙1= 3. When E 2 attains a slightly larger value,

432 2.5 2

|φ|.|ψ|

Fig. 1 A typical example of two components of a stable asymmetric soliton, with j.x/j  U ./, j .x/j  V ./, as per Sakaguchi and Malomed (2011). Continuous and dashed lines designate the numerically found solution and its VA-produced counterpart, respectively

B. A. Malomed

1.5 1 0.5 0 6

8

E22 D 6K;

10 x

12

14

(27)

a backward (subcritical) bifurcation (Iooss and Joseph 1980) occurs, which makes the symmetric solution with  D =4 unstable. The comparison with full numerical results corroborates the weakly subcritical shape of the SBB for solitons in the nonlinear coupler. A typical asymmetric soliton is displayed in Fig. 1, and the entire bifurcation diagram is presented in Fig. 2. Note that quantity cos.2/, which is used as the vertical coordinate in the diagram, measures the asymmetry of the soliton, because, as it follows from Eqs. (20) and (21), cos.2/ 

E .1/  E .2/ ; E .1/ C E .2/

(28)

where E .j / is the energy in the j -th core. Even without detailed stability analysis, one can easily distinguish between stable and unstable branches in the diagram, using elementary theorems of the bifurcation theory (Iooss and Joseph 1980). p predicts the backward bifurcation at the soliton’s energy E2 D p Thus, the VA 6K  2:45 K. The accuracy of the VA is characterized by comparison of this prediction with the abovementioned exactly found bifurcation value (17), the relative error being 0:057 (the analytical solutions for Ebif does not predict the subcritical character of the SBB). The above consideration addressed a single soliton in the nonlinear coupler. A cruder version of the VA was used in work (Doty et al. 1995) to analyze two-soliton interactions in the same system. Accurate numerical results for the interaction were reported in Peng et al. (1998). Furthermore, it was recently demonstrated that chains of stable solitons with opposite signs between adjacent ones, in the dual-core fiber, support the propagation of supersolitons, i.e., self-trapped collective excitations in the chain of solitons (Li et al. 2014). The underlying chain may be built of symmetric

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

433

1.0

cos(2θ)

0.5

0.0

–0.5

–1.0 0

1

2

3

4

5

E/ K

Fig. 2 The dependence of the asymmetry parameter of two-component p solitons in the nonlinear coupler with identical cores, cos.2 /, on the scaled total energy, E= K, as predicted by the VA, see Eq. (28). The figure demonstrates a weakly subcritical SBB, solid and dashed lines designating stable and unstable states, respectively. The results are presented as per Malomed et al. (1996b)

solitons, as well as of asymmetric ones, with alternating polarities, i.e., placements of larger and smaller components in the two cores.

Gap Solitons in Asymmetric Dual-Core Fibers Asymmetric dual-core fibers, consisting of two different cores, can be easily fabricated, and properties of solitons in them may be markedly different from those in the symmetric couplers. A general model of the asymmetric coupler is (cf. Eqs. (1), (2)) 1 i uz C qu C u  C juj2 u C v D 0; 2 

1 i vz  ı  qv C v  C jvj2 v C u D 0; 2

(29) (30)

where real parameter ı accounts for the difference between GVD coefficients in the cores, and another real coefficient, .1Cı/q, defines the phase-velocity mismatch between them, while a possible group-velocity mismatch can be eliminated in the equations by a simple transformation. The effect of the asymmetry between the cores on the SBB for solitons was addressed in Malomed et al. (1996b) and Kaup (1997) (strictly speaking, in this case the subject of the analysis is spontaneous breaking of quasi-symmetry, which remains after lifting the exact symmetry by the mismatch between the cores). In Kaup (1997), a VA-based analytical approach was elaborated, which showed good agreement with numerical results. A noteworthy feature of the SBB in the

434

B. A. Malomed

asymmetric model is a possibility of hysteresis in a broad region, while in the symmetric system, the hysteresis occurs in the narrow bistability region between the two bifurcation points, as seen in Fig. 2. A systematic analysis of the MI of CW states in the model of asymmetric nonlinear couplers was reported in Arjunan et al. (2017). The most interesting version of the asymmetric model is one with ı > 0 in Eq. (30), i.e., with opposite signs of the GVD (Kaup and Malomed 1998). In this case, the substitution of u; v  exp .i kz  i ! / in the linearized version of Eqs. (29) and (30) yields the respective dispersion relation,   1 k D .ı  1/ ! 2  2q ˙ 4

r

1 .ı C 1/2 .! 2  2q/2 C 1: 16

(31)

Self-trapped states may exist, as gap solitons (GSs), at values of the propagation constant, k, that belong to the gap in spectrum (31), i.e., such that values of ! corresponding to given k, as per Eq. (31), are unphysical (imaginary of complex). The gap always exists in the case of ı > 0, as seen from typical examples of the spectra for negative and positive mismatch q, which are displayed in Fig. 3. If the formal values of ! in the gap are complex, GS’s tails decay with oscillations, while for pure imaginary ! they decay monotonously. In particular, it follows from Eq. (31) that, in subgap 0 k 2 < 4ı= .1 C ı/2 , the tails always decay with oscillations. Solutions to Eqs. (29) and (30) for stationary GSs are sought for as u .z;  / D U ./ exp .i kz/, v .z;  / D V . / exp .i kz/, with real U and V determined by equations: .q  k/U C

1 d 2U C U 3 C V D 0; 2 d2

1 d 2V .ıq C k/V  ı 2 C V 3 C U D 0: 2 d

(32)

Approximate solutions to Eqs. (32) can be constructed by means of the VA, using the Gaussian ansatz     U D A exp  2 =2a2 ; V D B exp  2 =2b 2 :

(33)

Energies of the two components of the soliton corresponding to this ansatz are Z

C1

Eu  1

jU ./j2 dt D

p

A2 a; Ev 

Z

C1

jV ./j2 dt D

p 2 B b;

(34)

1

with the net energy E  Eu C Ev . Elimination of amplitudes A and B from the resulting system of variational equations leads to coupled equations for the widths a and b,

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers Fig. 3 Typical dispersion curves produced by Eq. (31) for the dual-core fiber with opposite signs of the GVD in the cores, corresponding to Eqs. (29) and (30) with ı D 1: (a) q D 1; (b) q D C1 (as per Kaup and Malomed 1998)

a

435

10

5

k

0

–5

–10 –4

b

–3

–2

–1

0

1

2

1

2

3

4

ω 6 4 2

k

0 –2 –4 –6 –4

–3

–2

–1

0

3

4

ω



  3ı C 4.k C ıq/b 2 D 32.ab/3 b 2  3a2  3  2 3b  a2 a2 C b 2 ; (35)         2 3  4.k  q/a2 3a2  b 2 a3 ı 3a2 C b 2 C 4.k C ıq/b 2 b 2  a2 D : b 3 Œ.3b 2 C a2 / C 4.k  q/a2 .b 2  a2 / Œ3ı C 4.k C ıq/b 2  .3b 2  a2 /2 3  4.k  q/a2



(36) These equations can be solved numerically, to find a and b as functions of propagation constant k and parameters ı and q. The results reported in Kaup and Malomed (1998) demonstrate that the GSs indeed exist in a part of the available gap, and, in most cases, they are stable. However, another part of the gap remains empty (there are intervals of k in the gap where no soliton can be found). A noteworthy feature of the GSs is that more than half of their net energy always resides in the normal-GVD component v, in spite of the obvious fact that the normal-GVD core cannot, by itself, support any

436 Fig. 4 A numerically found (solid lines) gap soliton solution of Eqs. (29) and (30) with oscillating decaying tails, and its VA-predicted counterpart (dashed lines), in the case of ı D 1 (opposite GVD in the two cores) and q D 0:2. The total energy of the numerically found gap soliton is E D 2:734 (as per Kaup and Malomed 1998)

B. A. Malomed 2.0 q=0.2, δ=1.0 U

U, V

0.0

V –2.0

0

1

2

3

4

5

6

7

τ

bright soliton. Further, a typical GS predicted by the VA (see Fig. 4) has a narrower component with a larger amplitude in the anomalous-GVD core, and a broader component with a smaller amplitude in the normal-GVD one, see Fig. 4. As is seen from Fig. 4, the VA generally correctly approximates the soliton’s core, but the simplest ansatz (33) does not take into regard the fact that, as mentioned above, the soliton’s tails decay with oscillations. The contribution of the tails also accounts for a conspicuous difference of the energy share Ev =E in the normal-GVD core from the value predicted by the VA for the same net energy E: for example, in the case shown in Fig. 4, the VA-predicted value is Ev =E D 0:585, while its numerically found counterpart is Ev =E D 0:516 (but it exceeds 1=2, as stressed above).

The Coupler with Separated Nonlinearity and Dispersion For better understanding of the light dynamics in strongly asymmetric nonlinear couplers, it is relevant to consider the model of an extremely asymmetric dual-core waveguide, in which the Kerr nonlinearity is carried by one core, and the GVD is concentrated in the other (Zafrany et al. 2005). Such a system, although looking “exotic,” can be created by means of available technologies, adjusting the zerodispersion point of the first core to the carrier wavelength of the optical signal and using a large effective cross-section area in the first core to suppress its nonlinearity. The respective system of coupled equations is i uz C juj2 u C v D 0;

(37)

i vz C qv C .D=2/v  C u D 0;

(38)

cf. Eqs. (1) and (2). Here the inter-core coupling coefficient is normalized to be 1, real parameter q, which may be positive or negative, is the phase-velocity mismatch between the cores, and D is the GVD coefficient, that we may be scaled to be C1 or 1, which corresponds to the anomalous or normal GVD, respectively. Group-velocity terms, such as i c1 u in Eq. (37) and i c2 v in Eq. (38), with some real

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

437

coefficients c1 and c2 , can be removed: the former one by the shift of the velocity of the references frame,  !   c1 z, and the latter one by the phase transformation, v ! v exp .i c2 =D/. Therefore, these terms are not included. Looking for a solution to the linearized version of Eqs. (37) and (38) in the usual form, fu; vg  exp .i kx  i !/ with real !, one arrives at the dispersion relation, 1 kD 2



 1 2 D!  q ˙ 2

s

1 4



1 D! 2  q 2

2 C 1:

(39)

Straightforward consideration of the spectrum defined by this expression demonstrates that, in the case of the anomalous GVD (D D C1), it gives rise to finite and semi-infinite gaps, 

  1 p 1 p 4 C q 2  q < k < 0I 4 C q 2 C q < k < 1; 2 2

(40)

and in the case of the normal GVD (D D 1), the semi-infinite and finite gaps are 1 0 and vice versa at q < 0.   Equation (39) can be inverted, to yield ! 2 D 2 .Dk/1 1 C qk  k 2 . This relation implies that, inside both the finite and semi-infinite gaps, ! 2 takes real negative values, suggesting a possibility to find exponentially localized solitons in both gaps. To realize this possibility, soliton solutions of Eqs. (37) and (38) were looked for as fu; vg D exp .i kz/ fU ./; V ./g. In the case of anomalous GVD, D D C1, it was thus found that the semi-infinite gap is completely filled by stable solitons, while the finite bandgap remains completely empty. This result does not depend on the magnitude and sign of the mismatch parameter, q, in Eq. (38). A typical example of the stable solitons found in the semi-infinite gap is shown in Fig. 5. Naturally, the shape of the soliton in the dispersive mode (V ) is much smoother than in the nonlinear one (U ). Nevertheless, the shapes of both components are strictly smooth; in particular, there is no true cusp at the tip of the dispersive one. The fact that the anomalous GVD supports stable solitons in the semi-infinite gap of the present system is not surprising, as the situation seems qualitatively similar to what is commonly known for the usual NLSE, even if the shape of the solitons is very different from that in the NLSE; see Fig. 5. More unexpected is the situation in the case of the normal GVD, D D 1 in Eq. (38). As found in Zafrany et al. (2005), in this case the semi-infinite gap remains empty, but the finite one is completely filled by stable GSs; see a typical example in Fig. 6. It is relevant to mention that Eqs. (37) and (38) are not Galilean invariant. In accordance with this, it was not possible to create moving solitons in the framework of this system.

438

a

0

–0.5

–1 U

Fig. 5 An example of a stable soliton generated by the two-core system with separated nonlinearity and dispersion, based on Eqs. (37) and (38), with parameters D D 1 (the anomalous sign of the GVD) and mismatch q D 0:8. The propagation constant corresponding to this soliton is k D 5, which places it in the semi-infinite gap, see Eq. (40). Panels (a) and (b) display, respectively, the nonlinear- and dispersive-mode components of the soliton

B. A. Malomed

–1.5

–2

–1

b

–0.5

0 τ

1

0.5

0 –0.01 –0.02

V

–0.03 –0.04 –0.05 –0.06 –0.07 –0.08 –3

–2

–1

0 τ

1

2

3

The analysis was also extended for the case when the nonlinear mode has weak residual GVD, of either sign (Zafrany et al. 2005). Still earlier, a similar model was considered in Atai and Malomed (2000), which introduced a two-core system with the nonlinearity in one core and a linear BG in the other. That system creates a rather complex spectral structure, featuring three bandgaps and a complex family of soliton solutions, including the so-called embedded solitons, which, under special conditions, may exist in (be embedded into) spectral bands filled by linear waves, where, generically, solitons cannot exist (Champneys et al. 2001).

Two Polarizations of Light in the Dual-Core Fiber A relevant extension of the model of the nonlinear coupler takes into regard two linear polarizations of light in each core. In this case, Eqs. (29) and (30) are replaced by a system of four equations Lakoba et al. (1997),

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

439

a 0.1 0 –0.1

U

–0.2 –0.3 –0.4 –0.5 –0.6 –1

–0.5

0 τ

b

0.5

1

0.03 0.025

V

0.02 0.015 0.01 0.005 0 –3

–2

–1

0 τ

1

2

3

Fig. 6 The same as in Fig. 5, but for parameters D D 1 (the normal sign of the GVD) and mismatch q D 0:8. The propagation constant of this stable soliton is k D 0:4, placing it in the finite bandgap; see Eq. (41)

i .u1 /z C i .v1 /z C i .u2 /z C i .v2 /z C

1 2 1 2 1 2 1 2

.u1 /  .v1 /  .u2 /  .v2 / 

C C C C

.ju1 j2 C .jv1 j2 C .ju2 j2 C .jv2 j2 C

2 jv j2 /u1 3 1 2 ju j2 /v1 3 1 2 jv j2 /u2 3 2 2 ju j2 /v2 3 2

C u2 D 0; C v2 D 0; C u1 D 0; C v1 D 0;

(42)

where fields u and v represent the two linear polarizations, the subscripts 1 and 2 label the cores, and the coupling coefficient is scaled to be K  1. In the case of

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B. A. Malomed

circular polarizations (rather than linear ones), the XPM (cross-phase-modulation) coefficient 2=3 in Eq. (42) is replaced by 2. Four-component soliton solutions to Eqs. (42) can be looked for by means of the VA based on the Gaussian ansatz:     u1;2 .z; / D A1;2 exp ipz  a2  2 =2 ; v1;2 .z; / D B1;2 exp i qz  b 2  2 =2 ; (43) with mutually independent real propagation constants p and q. Existence regions for all the solutions in the .p; q/ plane, produced by the VA for symmetric and asymmetric solitons (the asymmetry is again realized with respect to the two mutually symmetric cores), are displayed in Fig. 7, in the most essential case when the signs of amplitudes A1;2 and B1;2 in each polarization coincide (otherwise, all the solitons are unstable). Outside the shaded area in Fig. 7, there exist only solutions with a single polarization (i.e., with either v1;2 D 0 or u1;2 D 0), which were considered above. In particular, at the dashed-dotted borders of the shaded area, asymmetric four-component solitons (denoted by symbol AS1 in Fig. 7) carry over into the two-component asymmetric solitons of the single-polarization system. The symmetric solitons exist inside the sector bounded by straight continuous lines. The SBB, which gives rise to the asymmetric solitons AS1 and destabilizes the symmetric ones, takes place along the short-dashed curve in the left lower corner of the shaded area. There is an additional asymmetric soliton (AS2 in Fig. 7) in the inner area confined by the dashed curve. Thus, the total number of soliton solutions changes, as one crosses the bifurcation curves in Fig. 7 from left to right, from 1 to 3 to 5. However, soliton AS2 is generated from the symmetric one by an additional SBB, which takes place after the symmetric soliton has already been destabilized by the bifurcation that gives rise to asymmetric soliton AS1. For this reason, soliton AS2 is always unstable, while the primary asymmetric one AS1 is stable. Further details concerning the stability of different solitons in this model can be found in Lakoba and Kaup (1997).

Solitons in Linearly Coupled Fiber Bragg Gratings (BGs) In the systems described by the single or coupled NLSEs, the second-derivative terms account for the intrinsic GVD of the fiber or waveguide. On the contrary to this, strong artificial dispersion can be induced by a BG, i.e., a permanent periodic modulation of the refractive index written along the fiber (usually, the modulation is created in the fiber’s cladding), the modulation period being equal to half the wavelength of the propagating light. The nonlinear optical fiber carrying the BG is adequately described by the system of coupled-mode equations for amplitudes u.x; t / and v .x; t / of the right- and left-traveling waves (de Sterke and Sipe 1994; Aceves 2000):

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers Fig. 7 Regions of existence of the symmetric and two types of asymmetric (stable, AS1, and unstable, AS2) solitons in the plane (p; q) of two propagation constants of four-component solitons (43), in the model (42) of the dual-core-fiber system carrying two linear polarizations of light. Symbols u D 0, v D 0, and u D v refer to particular solutions with a single polarization and equal amplitudes of the two polarizations, respectively

a

441

8 7

u=0

u=v

6 AS1 AS2 5

q 4 v=0

AS1 3 2 1 0 0

2

6

4

8

p

b

8 7

u=0

6 5 AS

4

q 3 v=0 2 1 0 v=0 –1 –2 0

2

4

p

6

8

10

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B. A. Malomed

 i ut C i ux C .1=2/juj2 C jvj2 u C v D 0;  i vt  i vx C juj2 C .1=2/jvj2 v C u D 0:

(44) (45)

Here, the speed of light in the fiber’s material is scaled to be 1, as well as the linear-coupling constant, that accounts for mutual conversion of the right- and lefttraveling waves due to the resonant reflection of light on the BG. The ratio of the XPM and SPM (self-phase-modulation) coefficients in Eqs. (44) and (45), 2 W 1, is the usual feature of the Kerr nonlinearity. The dispersion relation of the linearized version of Eqs. (44) and (45) is ! 2 D 1C 2 k , hence the existence of GSs (alias BG solitons) with frequencies belonging to the corresponding spectral bandgap, 1 < ! < C1, may be expected. Indeed, although the system of Eqs. (44) and (45) is not integrable, it has a family of exact soliton solutions (Voloshchenko et al. 1981; Aceves and Wabnitz 1989; Christodoulides and Joseph 1989), which contains two nontrivial parameters, viz., amplitude Q; which takes values 0 < Q < , and velocity c, which belongs to interval 1 < c < C1. In particular, the solution for the quiescent solitons (c D 0) is p   uD p 2=3 .sin Q/ sech x sin Q  12 iQ  exp .i t cos Q/ ;   v D  2=3 .sin Q/ sech x sin Q C 12 iQ  exp .i t cos Q/ ;

(46)

where frequencies !sol  cos Q precisely fill the entire gap, while Q varies between 0 and . Stability of the BG solitons was investigated too, the result being that they are stable, roughly, in a half of the bandgap, namely, at 0 < Q < Qcr  1:01  .=2/ (Malomed and Tasgal 1994; Barashenkov et al. 1998; De Rossi et al. 1998). A natural generalization of the fiber BG is a system of two parallel-coupled cores with identical gratings written on both of them (Mak et al. 1998). The respective system of four coupled equations can be cast in the following normalized form, cf. Eqs. (44) and (45) for the single-core BG fiber and Eqs. (1) and (2) for the dual-core fiber without the BG:  1 ju1 j2 C jv1 j2 u1 C v1 C u2 2 

1 2 2 jv1 j C ju1 j v1 C u1 C v2 C 2 

1 ju2 j2 C jv2 j2 u2 C v2 C u1 C 2 

1 jv2 j2 C ju2 j2 v2 C u2 C v1 C 2

i u1t C i u1x C

D 0;

(47)

i v1t  i v1x

D 0;

(48)

D 0;

(49)

D 0;

(50)

i u2t C i u2x i v2t  i v2x

where  is the coefficient of the linear coupling between the two cores, which may be defined to be positive (unlike the models considered above, it is not possible to fix  D 1 by means of rescaling, because the scaling freedom has been already used

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

443

to fix the Bragg reflection coefficient equal to 1). The same model applies to the spatial-domain propagation in two parallel-coupled planar waveguides which carry BGs in the form of a system of parallel cores, in which case t and x play the roles of the propagation distance and transverse coordinate, respectively, while the paraxial diffraction in the waveguides is neglected. The dispersion relation for system (47), (48), (49), and (50) contains four branches (taking into regard that ! may have two opposite signs): p ! 2 D 2 C 1 C k 2 ˙ 2 1 C k 2 :

(51)

This spectrum has no gap in the case of strong inter-core coupling,  > 1. For the weaker coupling, with  < 1, the bandgap exists:  .1  / < ! < C .1  / :

(52)

To populate the bandgap, solutions for zero-velocity GSs are looked for as u1;2 D exp .i !t / U1;2 .x/ ; v1;2 D exp .i !t / V1;2 .x/ ;

(53)

 where relation V1;2 D U1;2 may be imposed (in fact, the exact GS solutions (46) in the single-core BG are subject to the same constraint). Substituting this in Eqs. (47), (48), (49), and (50) leads to two coupled equations (instead of four):

3 d U1 C juU1 j2 U1  U1 C U2 D 0 ; dx 2 3 d U2 C jU2 j2 U2  U2 C U1 D 0 : !U2 C i dx 2

!U1 C i

(54) (55)

Stationary equations (54) and (55) can be derived from their own Lagrangian, with density L D !.U1 U1 C U2 U2 / C

i 2



d U1  d U1 d U2  d U2 U1  U1 C U  U2 dx dx dx 2 dx

3 1 C .jU1 j4 C jU2 j4 /  .U12 C U12 C U22 C U22 / C .U1 U2 C U1 U2 / : 4 2



(56)

Then, in the framework of the VA the following ansatz may be adopted for the complex soliton solution sought for: U1;2 D A1;2 sech . x/ C iB1;2 sinh. x/ sech 2 . x/ ;

(57)

with real A1;2 , B1;2 , and . The integration of Lagrangian density (56) with this ansatz and subsequent application of the variational procedure give rise to the following system of equations:

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B. A. Malomed

3 2 3A2;1  3.1  !/A1;2 C 3A31;2 C A1;2 B1;2  B1;2 D 0 ; 5 3 3 3 B2;1 C B1;2  3: 857B1;2 C A21;2 B1;2  A1;2 D 0 ; 2 5

(58) (59)

 2!  2       B1 C B22 C A41 C A42  1: 285 7 B14 C B24 2! A21 C A22 C 3    2 2  4 2 B C B22 C 4A1 A2 C B1 B2 D 0 ; C A21 B12 C A22 B22  2 A21 C A22 C 5 3 1 3 (60) where numerical coefficients 3: 857 and 1:285 7 are defined by some integrals. A general result, following from both a numerical solution of variational equations (58), (59), and (60) and direct numerical solution of Eqs. (54) and (55), is that a symmetric mode, with A21 D A22 and B12 D B22 , exists at all values of ! in the bandgap (52), and it is the single soliton solution if the coupling constant  is close enough to 1, i.e., the bandgap (52) is narrow. However, below a critical value of  (which depends on given !), the symmetric solution undergoes a bifurcation, giving rise to three branches, one remaining symmetric, while two new ones, which are mirror images to each other, represent nontrivial asymmetric solutions. The bifurcation can be conveniently displayed in terms of an effective asymmetry parameter,  2   2  2 2 ‚  U1m = U1m C U2m ;  U2m

(61)

2 2 where U1m and U2m are peak powers (maxima of the squared absolute values) of complex fields U1;2 in the two cores (note its difference from the asymmetry parameter (28), which was defined in terms of integral energies, rather than peak powers). A complete plot of the SBB for the GSs in the present system, i.e., ‚ vs. ! and , is displayed in Fig. 8. At  D 0, when Eqs. (54) and (55) decouple, the numerical solution matches the exact solution (46) in one core, while the other core is empty. Note the difference of this supercritical (alias forward) SBB from its weakly subcritical (backward) counterpart for the solitons in the nonlinear coupler without the BG, which is shown in Fig. 2. The bifurcation diagram in Fig. 8 was drawn using numerical results obtained from the solution of Eqs. (54) and (55), but its variational counterpart is very close to it, a relative discrepancy between the VA-predicted and numerically exact values of , at which the SBB takes place for fixed !, being .5%. To directly illustrate the accuracy of the VA in the present case, comparison between typical shapes of a stable asymmetric soliton, as obtained from the full numerical solution and as predicted by the VA, is presented in Fig. 9. Direct numerical test of the stability of the symmetric and asymmetric solitons in the present model has yielded results exactly corroborating what may be expected:

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

445

1 0.8 0.6 θ 0.4 0.2 0 -0.2 0.8 0.6

0.5 ω

0.4 0

0.2

λ

-0.5

0

Fig. 8 The symmetry-breaking bifurcation diagram for zero-velocity gap solitons in the model of the dual-core nonlinear optical fiber with identical Bragg gratings written on both cores (as per Mak et al. 1998)

all the asymmetric solitons are stable whenever they exist, while all the symmetric solitons, whenever they coexist with the asymmetric ones, are unstable. However, all the symmetric solitons are stable prior to the bifurcation, where their asymmetric counterparts do not exist. Lastly, it is relevant to mention that influence of a possible phase shift between the BGs, written in the parallel-coupled cores, on four-component GSs in this system was studied too (Tsofe and Malomed 2007; Sun et al. 2013). In that case, the spontaneous (intrinsic) symmetry breaking is combined with the external symmetry breaking imposed by the mismatch between the BGs.

Bifurcation Loops for Solitons in Couplers with the Cubic-Quintic (CQ) Nonlinearity To conclude this section, it is relevant to briefly consider results obtained for the coupler with the CQ nonlinearity, i.e., a combination of competing self-focusing cubic and defocusing quintic terms in the respective system of coupled NLSEs: i uz C u  C 2juj2 u  juj4 u C v D 0; 2

4

i vz C v  C 2jvj v  jvj v C u D 0;

(62) (63)

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B. A. Malomed

0.8 0.6 0.4 0.2 0 -10

-8

-6

-4

-2

0 x

2

4

6

8

10

-8

-6

-4

-2

0 x

2

4

6

8

10

0.2 0.1 0 -0.1 -0.2 -10

Fig. 9 Shapes of the larger component of the quiescent soliton, U1 , in the dual-core Bragg grating (as per Mak et al. 1998). The upper and lower plots show the real and imaginary parts of U1 . Here, ! D 0:5 and  D 0:2. The soliton and dashed curves show numerical and variational results, respectively

cf. the usual system of Eqs. (1) and (2) with the cubic self-focusing (in both systems, the anomalous sign of the GVD is assumed). By means of straightforward rescaling, the coefficients in front of the nonlinear and dispersive terms may be fixed, without the loss of generality, as written in Eqs. (62) and (63), while coefficient  > 0 of the linear inter-core coupling remains a free irreducible parameter. Note that the PT symmetric version of the coupler with the CQ nonlinearity and solitons in it were considered too (Burlak et al. 2016). The CQ combination of the competing nonlinearities, which is assumed in the present system, occurs in various optical media. The realization which is directly relevant to the fabrication of dual-core fibers is provided by chalcogenide glasses (Smektala et al. 2000; Ogusu et al. 2004). The starting point of the analysis is a well-known exact soliton solution of the single CQ NLS equation (Pushkarov et al. 1979; Cowan et al. 1986), to which Eqs. (73) and (74) reduce in the symmetric case: u D v D e ikz Usymm . /;

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

v u u Usymm . / D t

1C

p

447

2 .k  /

 p ; 1  4 .k  / =3 cosh 2 k  

where the propagation constant k takes values in the interval of  < k < the limit cases of k D  and kD

3 C 4

(64)

3 4

C . In

(65)

this solution goes over, respectively, into the trivial zero solution and into the delocalized (continuous-wave, CW) state with a constant amplitude, u D v D q     3 exp i 34 C  z . The energy of soliton (64), which is defined by the same 2 expression (7) as above, is

Esymm

! p p p 3 3C2 k ln p D : p 2 32 k

(66)

Naturally, it diverges in the limit corresponding to Eq. (65). Following the pattern of the above analysis, asymmetric stationary soliton solutions to Eqs. (62) and (63) are looked for as fu.z; /; v.z; /g D e ikz fU ./; V ./g :

(67)

It can be proved (Albuch and Malomed 2007) that only solutions with real functions U ./ and V . / can be generated by the SBB from the symmetric soliton (64), hence the substitution of expressions (67) with real U and V in Eqs. (73) and (74) leads to a system d 2U  kU C V C 2U 3  U 5 D 0; d2

(68)

d 2V  kV C U C 2V 3  V 5 D 0; d2

(69)

which was solved numerically, and the stability of the so found solitons was then identified by dint of direct simulations (Albuch and Malomed 2007). The numerical solution of Eqs. (68) and (69) produces a sequence of bifurcation diagrams displayed in Figs. 10 and 11. In these diagrams, the soliton’s asymmetry parameter,



2 2  Vmax Umax ; 2 CV2 Umax max

(70)

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B. A. Malomed

Fig. 10 A set of bifurcation diagrams for symmetric and asymmetric solitons in the plane of the total energy, defined as in Eq. (7), but denoted N here (instead of E), and the asymmetry parameter (70). The diagrams are produced by numerical solution of Eqs. (12) and (13) with the cubic-quintic nonlinearity, at different values of the linear-coupling constant, . Stable and unstable branches of the solutions are shown by solid and dashed curves, respectively, and bold dots indicate bifurcation points (as per Albuch and Malomed 2007)

2 2 where Umax and Vmax are the peak powers of the two components of the soliton, is shown versus its total energy. Note the similarity of this definition of the asymmetry to that adopted above in the form of Eq. (61) for solitons in the dual-core BG. A remarkable peculiarity of the present system is the existence of the bifurcation loop: as Figs. 10 and 11 demonstrate, the direct SBB, which occurs with the increase of the energy, being driven, as above, by the cubic self-focusing, is followed, at larger energies, by the reverse bifurcation, which takes place when the dominant nonlinearity becomes self-defocusing, represented by the quintic terms in (68) and (69). The loop exists at 0 <  max  0:44. The direct bifurcation is seen to be always supercritical, while the reverse one, which closes the loop, is subcritical (giving rise to the bistability and concave shape of the loop, on its right-hand side) up to   0:40. In the interval of 0:40 <  < 0:44, the reverse bifurcation is supercritical, and the (small) loop has a convex form. The picture of the bifurcations is additionally illustrated by Fig. 12, which displays the energy of the symmetric soliton at points of the direct and reverse bifurcations.

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers lamda=0.32 0.8

0.4

0.4

epsilon

epsilon

lamda=0.28 0.8

0 -0.4 -0.8

0

1

449

2

3

0 -0.4 -0.8

3.5

0

1

N lamda=0.36

0.4 0 -0.4 -0.8

0

1

2

3

0

1

epsilon

epsilon

0.8

0 -0.4 1

3

3.5

2

3

3.5

N

lamda=0.43

0

2

0 -0.4 -0.8

3.5

0.4

-0.8

3.5

0.4

N 0.8

3

lamda=0.4

0.8

epsilon

epsilon

0.8

2 N

2

3

3.5

lamda=0.438

0.4 0 -0.4 -0.8

0

1

N

N

Fig. 11 Continuation of Fig. 10 to larger values of the coupling constant, 

Stability and instability of different branches of the soliton solutions can be anticipated on the basis of general principles of the bifurcation theory (Iooss and Joseph 1980): the symmetric solution becomes unstable after the direct supercritical bifurcation, and asymmetric solutions emerge as stable ones at this point; eventually, the reverse bifurcation restores the stability of the symmetric solution. In the case when the reverse bifurcation is subcritical and, accordingly, the bifurcation loop is concave on its right side, two branches of asymmetric solutions meet at the turning points, the branches which originate from the reverse-bifurcation point being unstable. These expectations are fully borne out by direct numerical simulations (Albuch and Malomed 2007). In particular, in the case when the bifurcation loop has the concave shape, an unstable asymmetric soliton has a choice to evolve into either a still more asymmetric one or the symmetric soliton (also stable). Numerical results clearly demonstrate that unstable asymmetric solitons choose the former option, evolving into the more asymmetric counterparts.

450

B. A. Malomed 4.5 Nfirst bifurcation Nsecond bifurcation

4 3.5

N

3 2.5 2 1.5 1 0.5

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

lamda

Fig. 12 Values of the energy of the symmetric soliton at which the direct and reverse bifurcations occur in Figs. 10 and 11. The two curves merge and terminate at  D max  0:44

Dissipative Solitons in Dual-Core Fiber Lasers Introduction Experimental and theoretical studies of fiber lasers are, arguably, the fastest developing area of the modern laser science (Richardson et al. 2010). A commonly adopted model for the evolution of optical pulses in fiber lasers is based on complex Ginzburg-Landau equations (CGLEs), which readily predict formation of dissipative solitons, alias solitary pulses (SPs), in the lasers (Grelu and Akhmediev 2012), due to the stable self-sustained balance of loss and gain, the latter provided by the lasing mechanism (typically, stimulated emission of photons by externally pumped ions of rear-earth metals which are embedded as dopants into the fiber’s silica (Richardson et al. 2010)). Important applications of the CGLEs are known in many other fields, including hydrodynamics, plasmas, reaction-diffusion systems, etc., as well as other areas of nonlinear optics (Aranson and Kramer 2002; Malomed 2005). The CGLE of the simplest type is one with the linear dispersive gain and cubic loss (which represents two-photon absorption), combined with the GVD and Kerr nonlinearity. This equation readily produces an exact analytical solution for SPs, in the form of the chirped hyperbolic secant (Hocking and Stewartson 1972; Pereira and Stenflo 1977), but they are unstable, for an obvious reason: the linear gain destabilizes the zero background around the SP. The most straightforward modification which makes the existence of stable SPs possible is the introduction

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

451

of the cubic-quintic (CQ) nonlinearity, which includes linear loss (hence the zero background is stable), cubic gain (provided by a combination of the usual linear gain and saturable absorption), and additional quintic loss that provides for the overall stabilization of the model. The CGLE with the CQ nonlinearity was first proposed (in a 2D form) by Sergeev and Petviashvili (Petviashvili and Sergeev 1984). A stable SP solution in the 1D version of this equation, which is relevant to modeling fiber lasers, was first reported, in an approximate analytical form, in Malomed (1987b). These solutions were found by treating the dissipation and gain terms in the CGLE as small perturbations added to the usual cubic NLSE with the anomalous sign of the GVD. Accordingly, the SP was obtained as a perturbation of the standard NLSE soliton. In later works, SPs and their stability in the CQ CGLE were investigated in a broad region of parameters (van Saarloos and Hohenberg 1990; Malomed and Nepomnyashchy 1990; Hakim et al. 1990; Marcq et al. 1994; Soto-Crespo et al. 1996). Another possibility to produce stable SPs, which is directly relevant to the general topic of the present chapter, is to linearly couple the usual cubic CGLE to an additional equation which is dominated by the linear loss. A coupled system of this type was first introduced in Malomed and Winful (1996), as a model of a dualcore nonlinear dispersive optical fiber, with linear gain, 0 , in one (active) core, and linear loss, 0 , in the other (passive) one: 1 i uz C u  C juj2 u  i 0 u  i 1 u  C v D 0; 2

(71)

i vz C .1=2/v  C jvj2 v C i 0 v C u D 0:

(72)

Here, u and v are, respectively, envelopes of the electromagnetic waves in the active and passive cores, z and  are, as above, the propagation distance and reduced time, cf. Eqs. (1) and (2),  is the constant of the inter-core coupling, and 1 accounts for dispersive loss in the active core (in other words, 1 determines the bandwidthlimited character of the linear gain). The model assumes the usual self-focusing Kerr nonlinearity and anomalous GVD in the fiber, with the respective coefficients scaled to be 1. The gain in the dual-core fiber can be experimentally realized, similar to the usual fiber lasers, by means of externally pumped resonant dopants (Li et al. 2005). Actually, the dual-core fiber may be fabricated as a symmetric one, with both cores doped, while only one core is pumped by an external light source, which gives rise to the gain in that core. A more general system of linearly coupled CGLEs applies to a system of parallel-coupled plasmonic waveguides (Marini et al. 2011). The system was further extended for coupled 2D CGLEs, representing stabilized laser cavities (Firth and Paulau 2010; Paulau et al. 2010, 2011). It was first theoretically predicted in Winful and Walton (1992) and Walton and Winful (1993) that, adding the parallel-coupled passive core (with loose ends) to the soliton-generating fiber laser, one can improve the stability of the output: while the soliton, being a self-trapped nonlinear mode, remains essentially confined to

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B. A. Malomed

the active core, small-amplitude noise easily couples to the passive one, where it is radiated away through the loose ends. Independently, a similar dual-core system, with loss in the additional core but without gain in the main one, was proposed as an optical filter cleaning solitons from noise (Chu et al. 1995a). The basic idea is that, due to the action of the self-focusing nonlinearity, the soliton keeps itself in the core in which it is propagating, while the linear noise tunnels into the parallel core, where it is suppressed by the loss. It was found that the best efficiency of the filtering is attained not with very strong loss ( 0 ) in the extra core, but rather at 0  ; see Eq. (72) (Chu et al. 1995a). Another possible application of the dualcore system with the gain in the straight core (the one into which the input signal is coupled) and losses in the cross core (the one linearly coupled to the straight core) was proposed for the design of a nonlinear amplifier of optical signals: a very weak (linear) input would pass into the cross core and would be lost there, while the input whose power exceeds a certain threshold, making it a sufficiently nonlinear mode, stays in the straight core, being amplified there (Malomed et al. 1996a). In Malomed and Winful (1996), the possibility of the existence of stable SPs in the system of Eqs. (71) and (72) was predicted in the framework of an analytical approximation that treated both the coupling and gain/loss terms as small perturbations. The stability of the so predicted pulses was then verified by direct simulations (Atai and Malomed 1996). Further, it was demonstrated that the system may be simplified, by dropping the nonlinear and GVD terms in Eq. (72), where they are insignificant (the nonlinearity may be omitted as the amplitude of the component in the passive core is small, and the GVD is negligible, as the linear properties of the passive core are dominated by the loss term). On the other hand, an extra linear term, namely, a phase-velocity mismatch between the cores, should be added to Eq. (72), as the respective effect may be essential (see below). As a result, a pair of SP solutions for the simplified system was found in an exact analytical form (which is displayed below), the pulse with a larger amplitude being stable in a vast parameter region, while its counterpart with the smaller amplitude is always unstable (Atai and Malomed 1998b; Efremidis et al. 2000a). The basic results are presented, in some detail, in subsections following below. A detailed review of these and related results can be found in Malomed (2007).

The Exact SP (Solitary-Pulse) Solution The most fundamental coupled system, which has the cubic nonlinearity in the active core only, is based on the following system, which is simplified in comparison with original equations (71) and (72) (Atai and Malomed (1998b); see also MartiPanameno et al. (2001)).

i uz C

 1  i 1 u  C . C i 2 / juj2 u  i 0 u C v D 0; 2 i vz C k0 v C i 0 v C u D 0;

(73) (74)

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

453

where  D 1 is fixed by means of scaling, D C1 and 1 correspond to the anomalous and normal GVD in the active core (assuming that the actual nonlinearity is self-focusing, which is the case in optical fibers, this sign parameter may be placed in front of the cubic term, as written in Eq. (73), although originally appears in front of the second derivative), 2 0 accounts for cubic loss (two-photon absorption), and k0 is the abovementioned phase-velocity mismatch between the cores. The exact SP solution to Eqs. (73) and (74) can be found in the analytical form suggested by the well-known solution (Hocking and Stewartson 1972; Pereira and Stenflo 1977) of the cubic CGLE: fu; vg D fA; Bg e ikz Œsech ./1Ci ;

(75)

where all the constants but B are real. Coefficient , which determines the chirp of the pulse, is

D

q 9 .1  2 1 2 /2 C 8 .21 C 2 /2  3 .1  2 1 2 /

:

(76)

B D .k  k0  i 0 /1 A;    .1  2 1 2 / 2  2 C 3 .21 C 2 / 2 2   A D ; 2 1 C 222

(77)

2 .21 C 2 /

The complex and real amplitudes, B and A, are given by expressions

(78)

with the two remaining real parameters  and k determined by one complex equation, k C i 0  .k  k0  i 0 /

1

D

 1  i 1 .1 C i /2 2 : 2

(79)

In the further analysis of the SP solutions, one may set 2 D 0, as the two-photon absorption is insignificant in silica fibers. Then, 2 can be eliminated from Eq. (79), 0 1  8 0 1 0 @1  h iA ; 2 D q 2 2 2 2 .k /  k  C 81 C 3  9 C 321 0 0 0

(80)

and one arrives at a final cubic equation for k: i h

k .k  k0 /2 C 02  .k  k0 / D

q i 9 C 3212 h 0 .k  k0 /2  0 .1  0 0 / ; 21 (81)

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B. A. Malomed

which may give rise to one or three real solutions. Physical solutions are those which make expression (80) positive. In particular, the number of the physical solutions changes when expression (80) vanishes, which happens at k02 D .0 0 /1 .1  0 0 / . 0  0 /2 . The above results may be cast in a more explicit form in the case of no wavenumber mismatch between the cores, k0 D 0. Note that the SP solution is of interest if it is stable, a necessary condition for which is the stability of the zero background, i.e., the trivial solution, u D v D 0. If k0 D 0, necessary conditions for the stability of the zero solution are 0 < 0 and 0 0 < 1. It is natural to focus on the case when gain 0 is close to the maximum value, .0 /max  1= 0 , admitted by the latter condition, i.e., 0 < 1  0 0  1

(82)

(then, condition 0 < 0 reduces to 0 > 1). In this case, Eq. (81) may be easily solved. The first root has small k, which yields unphysical solutions, with 2 < 0. Two other roots for k are physically relevant ones, in which 0 may be replaced by 1= 0 , due to relation (82):

q 9 C 3212

s

  9 C 3212  16 02  1 ; 2 2 0 1 0 1   1 81 k 2 k 2 C 02

: 2 D q 0 3 C 812  9 C 3212

4k D

˙

(83)

(84)

Expression (84) is always positive, while a nontrivial existence condition for these two solutions follows from Eq. (83): k must be real, which means that   912 C 32 > 16 02 02  1 :

(85)

Thus, Eqs. (83) and (84), along with Eqs. (75), (76), (77), (78), (80), and (81), furnish the SP solutions in the region of the major interest, and Eq. (85) is a fundamental condition which secures the existence of these solutions. If condition (85) holds, one has the following set of solutions: (i) the stable zero state, (ii) the broader SP with a smaller amplitude, corresponding to smaller k 2 , i.e., with sign ˙ in expression (83) chosen opposite to , and (iii) the narrower pulse with a larger amplitude, corresponding to larger k 2 , i.e., with ˙ in (83) chosen to coincide with . Basic principles of the bifurcation theory (Iooss and Joseph 1980) suggest that stable and unstable solutions alternate, hence, because the trivial solution is stable, the larger-amplitude narrower pulse ought to be stable too, while the intermediate broader pulse with the smaller amplitude is always unstable, playing the role of a separatrix between the two attractors. This expectation is, generally, corroborated by numerical results (Atai and Malomed 1996, 1998b), as

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

455

shown in some detail below. Note that the abovementioned SP waveform (Hocking and Stewartson 1972; Pereira and Stenflo 1977), which suggested ansatz (75) for the exact solutions under the consideration, is, by itself, always unstable as the solution of the single cubic CGL equation.

Special Cases of Stable SPs (Solitary Pulses) There are two particular cases of physical interest that should be considered separately. The first corresponds to the model with 1 D 0 (negligible dispersive loss). In this case, the above SP solution may only exist in the system with anomalous GVD, D C1. Special consideration of this case is necessary because the above formulas are singular for 1 D 0. An explicit result for this situation can be obtained without adopting condition (82): the solutions take the form of Eq. (75) with D 0 (no chirp), Eq. (78) being replaced by A2 D 2 , while Eqs. (79) and (78) become q k  k0 D ˙ 0 01 .1  0 0 /; 2 D 2k0 ˙ 2 .0 C 0 /

q .0 0 /1 .1  0 0 /:

(86)

Two solutions corresponding to both signs in Eqs. (86) exist simultaneously, i.e., 2 > 0 holds for both of them, provided that k02 > .0 0 /1 .1  0 0 / .0 C 0 /2 . However, one can check that this inequality contradicts the stability conditions of the zero state, therefore only one solution given by Eqs. (86) may exist in the case of interest. General principles of the bifurcation theory (Iooss and Joseph 1980) suggest that this single nontrivial solution is automatically unstable in the case of 1 D 0, once the trivial one is stable. The other specially interesting case is that of zero GVD, corresponding to the physically important situation when the carrier wavelength is close to the zerodispersion point (Agrawal 2007) of the optical fiber. In this case, Eq. (74) does not change its form, while Eq. (73) is replaced by i uz  i u  C juj2 u  i 0 u C v D 0

(87)

(in the absence of the GVD, both signs of are equivalent, hence D C1 is fixed here, and normalization 1  1 may be adopted). Explicit solutions can be obtained, as well as in the general case considered above, by assuming k0 Dp0 and taking p0 0 close to 1. Then, expressions (78) and (76) are replaced by D 2, A2 D 3 22 , while solutions (83) and (84) take the form of kD

p 1 q  1 2 2 0 ˙ 2 02  02 C 1; 2 D 01 02 C k 2 k ;

(88)

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B. A. Malomed

and the existence condition (85) becomes very simple, 02 < 2. Thus, in the zeroGVD case, both SP solutions (88) exist simultaneously, suggesting that the one with the larger value of k 2 may be stable. Lastly, it may be interesting to consider another particular case, with normal GVD, D 1, and small dispersive-loss coefficient, 1  1. In this case, there are two nontrivial SP solutions, the one with the larger amplitude that has the chance to be stable, being 3 3 3 4  1  1 : D  11 ; A2 D .1 0 /1 ; k D  11 ; 2 D 2 2 2 3 0

(89)

The large value of in this solution implies that the pulse is strongly chirped. Obviously, the latter solution disappears in the limit of 1 ! 0, in accordance with the abovementioned negative result (no stable SP) for the case of 1 D 0.

Stability of the Solitary Pulses and Dynamical Effects As mentioned above, the SP cannot be stable unless its background, u D v D 0, is stable. To explore the stability of the zero solution, infinitesimal perturbations are substituted in the linearized version of Eqs. (73) and (74), fu; vg D fu1 ; v1 g e i.qz!t/ , where ! and q are an arbitrary real frequency and the corresponding propagation constant (generally speaking, a complex one). The stability condition is Im.q/ D 0, which must hold at all real !. This condition leads an inequality that should be valid at all ! 2 0, " 2

0 .0  1 ! / 1 C

 2 2k0 C ! 2 4 . 0  0 C 1 ! 2 /2

# 1:

(90)

If the GVD coefficient is zero, i.e., Eq. (73) is replaced by Eq. (87), expression  2 2k0 C ! 2 in Eq. (90) is replaced by 4k02 . An obvious corollary of Eq. (90) is 0 < 0 ;

(91)

i.e., the trivial solution may be stable only if the loss in the passive core is stronger than the gain in the active one. Further, in the case of zero wavenumber mismatch between the cores, k0 D 0, which was considered above (see Eqs. (83) and (84)), a simple necessary stability condition is obtained from Eq. (90) at ! D 0 (as was already mentioned above, see Eq. (82)): 0 0 > 1. Stability regions of the SP solutions in the parameter space of the system were identified in Atai and Malomed (1996) and Efremidis et al. (2000a) in a numerical form, combining the analysis of the necessary stability condition of the zero background, given by Eq. (90), and direct simulations of Eqs. (73) and (74) for perturbed SPs. As said above, the case of major interest is the one with k0 D 2 D 0

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers Fig. 13 The stability region (shaded) in the parameter plane of the exact solitary-pulse solution (75) of Eqs. (73) and (74), for the case of k0 D 2 D 0 and 0 0 D 0:9, as per Atai and Malomed (1998b) and Malomed (2007). Panels (a) and (b) display results for the anomalous and normal GVD, respectively, i.e., D C1 and D 1 in Eq. (73). The separate curve shows the existence boundary given by Eq. (85)

a

457

1.0

0.8

0.6

γ0 0.4

0.2

0.0 0.0

b

1.0

2.0

1.0

2.0

γ1

1.0

0.8

0.6

γ0 0.4

0.2

0.0 0.0

γ1

and 0 0 close to 1. For this case, the stability regions are displayed in Fig. 13 and separately in Fig. 14 for the zero-GVD system. In each case, there is a single stable SP (but the system is a bistable one, as the stable SP always coexists with the stable zero solution). Another cross section of the stability region in the full three-dimensional parameter space of the model is represented by region II in Fig. 15, for the normalGVD case ( D 1), with 1 D 1=18 (this value is a typical one corresponding to physically relevant parameters of optical fibers). Note that this plot reveals a very narrow region (area III), in which the zero solution is stable, while the SP is not.

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B. A. Malomed

Fig. 14 The same as in Fig. 13, but for the exact solitary-pulse solution (88), in the system with zero dispersion (as per Atai and Malomed 1998b and Malomed 2007)

1.6

1.2

Γ0

0.8

0.4

0.0 0.0

0.2

0.4

γ0

0.6

0.8

1.0

15

Γ

10

IV 5 II III I 0

0

5

10 K

15

20

Fig. 15 Stability regions of the zero solution and existence and stability regions for the exact solitary-pulse solution (75) of Eqs. (73) and (74), as per Atai and Malomed (1998b) and Malomed (2007), in the plane of parameters K  1=0 and  0 =0 , in the model with the normal GVD . D 1), k0 D 2 D 0, and 1 D 1=18. Region I: the zero background is unstable. Region II: the solitary pulse is stable. Region III: the zero background is stable, while the solitary pulse is not, decaying to zero in direct simulations. Region IV: the solitary-pulse solution does not exist. In the region located outside of region I, which is bordered by curves 0 0 D 1 (the dotted curve) and 0 D 0 (the dotted-dashed curve), the zero solution is certainly unstable, as condition (82) does not hold in that region. In region I, zero background is unstable even though condition (82) holds in this region

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

a 1000 750 500 40 30 2 20 | |u 10

250

0

–100

0

100

Normalized Distance z

Fig. 16 Merger of chirped solitary pulses in the system of Eqs. (73) and (74) with normal GVD ( D 1). Other parameters are k0 D 2 D 0, 0 D 0:2, 0 D 0:8, and 1 D 1=18. The initial phase shift between the pulses is  D 0 (a) and  D  (b). The results are displayed as per Atai and Malomed (1998a) and Malomed (2007)

459

0 200

Normalized Time t

b

750 500 40 250

30 20 2

| |u

10

0

–100

0

100

Normalized Distance z

1000

0 200

Normalized Time t

Interactions Between Solitary Pulses It is well known that the sign of the interaction between ordinary solitons is determined by the phase shift between them, : the interaction is attractive for in-phase soliton pairs, with  D 0, and repulsive for  D  (Kivshar and Malomed 1989a). However, this rule does not apply to the SPs in the present model with the normal sign of the GVD ( D 1 in Eq. (73)), which feature strong chirp in their phase structure (see, e.g., the expression for chirp in solution (89), with 1  1). It was found (Efremidis et al. 2000a) that, irrespective of the value of , the SPs in the normal-GVD model attract each other and eventually merge into a single pulse, as shown in Fig. 16. On the other hand, in-phase pairs of the SPs in the anomalous-GVD system, with D C1, readily form robust bound states (Atai and Malomed 1998a). Threepulse bound states were found too, but they are unstable against symmetry-breaking perturbations, which split them according to the scheme 3 ! 2 C 1.

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B. A. Malomed

CW (Continuous-Wave) States and Dark Solitons (“Holes”) In addition to the SPs, Eqs. (73) and (74) also admit CW states with constant amplitudes, u D a exp .i kz  i !/ ; v D b exp .i kz  i ! / :

(92)

The propagation constant and amplitudes of this solution can be easily found in the case of 2 D 0: b D .k  k0  i 0 /1 a; q k  k0 D ˙ 0 Q01 .1  0 Q0 /;

a 2 D k0 ˙

q . 0 Q0 /1 .1  0 Q0 / . 0  Q0 / ;

(93)

where Q0  0  1 ! 2 . According to this, at given ! there may exist two CW states with different amplitudes, provided that k02 . 0 Q0 /1 .1  0 Q0 / . 0  Q0 /2 , and a single one in the opposite case. The CWs may be subject to the MI (modulational instability), which was investigated in detail (Afanasjev et al. 1997; Ganapathy et al. 2006). In particular, all CWs are unstable at k0 D 0, although the character of the MI is different for the normal and anomalous signs of the GVD (Ganapathy et al. 2006). Direct simulations demonstrate that the development of the MI splits the CW state into an array of SPs. At k0 ¤ 0, there is a parameter region in the normal-GVD regime (with D 1) where the CW solutions are stable, which suggests to explore solutions in the form of dark solitons (which are frequently called “holes,” in the context of the CGLEs (Nozaki and Bekki 1984; Sakaguchi 1991)). Such solutions of Eqs. (73) and (74) can be found in an exact analytical form based on the following ansatz (Efremidis et al. 2000b) (cf. the form of exact solution (75) for the bright SP):

uD

  2  e 2 A .1 C

e 2 /1i

e ikzi  ; v D

u ; k0  k C i 0

(94)

q with D .3=41 / C .9=41 /2 C 2 (in the case of 2 D 0), the other parameters, A; , and k, being determined by cumbersome algebraic equations. Numerical analysis demonstrates that dark solitons (94) are stable in a small parameter region, as shown in Fig. 17.

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

461

10

8

Γ

6

III

4

IV

I 2

II

0 0

2

4

6

8

10

K Fig. 17 Regions of the existence and stability of the CW state and dark soliton (94), produced by Eqs. (73) and (74) in the same parameter plane as in Fig. 15, in the case of D 1, 2 D 0, 1 D 1=7, and k0 D 22 . The existence region of the CW solution is confined by the dashed lines, 0 D 1 and 0 0 D 1. In regions I and IV, the CW is unstable. In region II, it is stable, while the dark soliton is not. In region III, both the CW and dark-soliton solutions are stable. The results are displayed as per Efremidis et al. (2000a) and Malomed (2007)

Evolution of Solitary Pulses Beyond the Onset of Instability Direct simulations of the system with normal GVD, D 1 in Eq. (74), demonstrate that unstable SPs (in the case when stable solutions do not exist) either decay to zero (if the zero background is stable) or blow up, initiating a transition to a “turbulent” state, if the background is unstable. A different behavior of unstable SPs was found in Sakaguchi and Malomed (2000) in the model with the anomalous GVD ( D C1). If the instability of the zero background is weak, it does not necessarily lead to the blowup. Instead, it may generate a small-amplitude background field featuring regular oscillations. In that case, the SP sitting on top of such a small-amplitude background remains completely stable, as shown in Fig. 18. The proximity of this state to the stability border is characterized by the overcriticality parameter,

 .0  .0 /cr / = .0 /cr ;

(95)

where .0 /cr is the critical size of the linear gain at which the instability of the zero solution sets in, at given values of 1 , k0 , and 0 . An example of the stable pulse

462

B. A. Malomed 5 IuI 4

3

2

1

0

20

0

40

60 x

80

100

120

Fig. 18 An example of a stable solitary pulse in the system of Eqs. (73) and (74), with the anomalous GVD ( D C1), which exists on top of the small-amplitude background featuring regular oscillations, in the case when the zero background is weakly unstable (the respective overcriticality is D 0:025; see Eq. (95)). Parameters are k0 D 0, 0 D 0:54, 0 D 1:35, and 1 D 0:18. The results are displayed as per Sakaguchi and Malomed (2000) and Malomed (2007)

a

b

6

1600 1400

IuI 5

1200 4

1000 t

3

800 600

2

400 1 0

200 0

20

40

60 x

80

100

120

0

0

20

40

60 x

80

100 120

Fig. 19 (a) An example of a solitary pulse which remains stable, as a whole, on top of the background featuring chaotic oscillations, at overcriticality D 0:157. Parameters are the same as in Fig. 18, except for 0 D 0:61: (b) The random walk of the stable pulse from (a), driven by its interaction with the chaotic background. The results are displayed as per Sakaguchi and Malomed (2000) and Malomed (2007)

found on top of the finite background, which is displayed in Fig. 18, pertains to

D 0:025. At larger but still moderate values of the overcriticality, such as D 0:157 in Fig. 19, the background oscillations become chaotic, while keeping a relatively small amplitude. As a result of the interaction with this chaotic background, the

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

463

SP remains stable as a whole, featuring a random walk. The walk shows a typically diffusive behavior, with the average squared shift in the  -direction growing linearly with z (Sakaguchi and Malomed 2000). The randomly walking pulses may easily form bound states, which then feature synchronized random motion. Finally, at essentially larger values of the overcriticality, & 1, the system goes over into a turbulent state, which may be interpreted as a chaotic gas of solitary pulses (Sakaguchi and Malomed 2000).

Soliton Stability in PT (Parity-Time)-Symmetric Nonlinear Dual-Core Fibers The dual-core fibers with equal (mutually balanced) gain and loss in the cores offer a natural setting for the realization of the PT symmetry, in addition to other optical media, where this symmetry was proposed theoretically (Ruschhaupt et al. 2005; El-Ganainy et al. 2007; Musslimani et al. 2008; Makris et al. 2008, 2011; Klaiman et al. 2008; Longhi 2009; Suchkov et al. 2016; Konotop et al. 2016; Burlak et al. 2016) and implemented experimentally (Guo et al. 2009; Ruter et al. 2010). The basic model of the PT -symmetric nonlinear coupler is based on the equations similar to Eqs. (1) and (2), in which  > 0 is the gain-loss coefficient, and the coefficient of the inter-core coupling (K in Eqs. (1) and (2)) is scaled to be 1 (Driben and Malomed 2011a; Alexeeva et al. 2012): i uz C .1=2/ut t C juj2 u  i  u C v D 0;

(96)

i vz C .1=2/vt t C jvj2 v C i  v C u D 0:

(97)

Note that the PT -balanced gain and loss in this system correspond to the border between stable and unstable settings: if the loss coefficient in Eq. (97) is replaced by an independent one, > 0 (different from the gain factor  in Eq. (97)), the zero solution, u D v D 0, is unstable at  > and may be stable at  < ; see Eq. (91). Obviously, p any solution to the NLSE (with a frequency shift), i Uz C .1=2/Ut t C jU j2 U ˙ 1   2 U D 0, gives rise to two exact solutions of the PT -symmetric system, provided that condition  1 holds:   p v D i  ˙ 1   2 u D U .z; t / :

(98)

For  D 0, solutions (98) with C and  amount, respectively, to symmetric and antisymmetric modes in the dual-core coupler, therefore the respective solutions (98) may be called PT -symmetric and PT -antisymmetric ones. In the limit of  D 1, two solutions (98) reduce to a single one, v D i u D U .z; t /. In particular, PT symmetric and antisymmetric solitons, with arbitrary amplitude , are generated by the NLSE solitons,

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B. A. Malomed

    p U .z; t / D  exp i 2 =2 ˙ 1   2 z sech .t / :

(99)

As concerns stability of the solitons in this system, it is relevant to compare it to the stability in the usual coupler model, with  D 0. As explained above (see Eq. (17)), the symmetric solitons in the nonlinear coupler are unstable against the spontaneous symmetry breaking at 2 > 2max . D 0/  4=3

(100)

(Wright et al. 1989), while antisymmetric solitons are always unstable (Soto-Crespo and Akhmediev 1993) (although their instability may be weak). The analysis of the instability of the usual two-component symmetric solitons against antisymmetric perturbations, ıu D ıv, which leads to the exact result (100) (see Eqs. (14) and (16), can be extended for the PT -symmetric system. The respective perturbation ıu at the critical point, 2 D 2max , obeys the linearized equation, n p  o 4 1   2  d 2 =dt 2 C 2max 1  6sech2 .max t / ıu D 0;

(101)

which is the respective generalization of Eq. (16). This equation with the PöschlTeller potential is solvable, yielding p 2max . / D .4=3/ 1   2 :

(102)

This analytical prediction was verified by direct simulations of the perturbed evolution of the PT -symmetric solitons. The simulations were run by adding finite initial antisymmetric perturbations, at the amplitude level of ˙3%, to the symmetric solitons. For PT -antisymmetric solitons, the instability boundary was identified solely in the numerical form, by running the simulations with initial symmetric perturbations. The results are summarized in Fig. 20, as per Driben and Malomed (2011a). The numerically identified stability border for the symmetric solitons goes somewhat below the analytical one (102) because the finite perturbations used in the simulations are actually not quite small. Taking smaller perturbations, one can obtain the numerical stability border approaching the analytical limit. For instance, at  D 0:5, the perturbations with relative amplitudes ˙5%, ˙3%, and ˙1% give rise to the stability border at 2max D 1:02, 1:055, and 1:08, respectively, while Eq. (102) yields 2max  1:15 in the same case. As concerns the PT antisymmetric solitons, a detailed analysis demonstrated that they are completely unstable (Alexeeva et al. 2012), while the numerically found boundary delineates an area in which the instability is very weak. It is relevant to stress that, being the stability boundary of the PT -symmetric solitons, the present system, unlike the usual dual-core coupler (see above), cannot support asymmetric solitons, as the balance between the gain and loss is obviously

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

465

1.5

η2max

1

0.5

0

Symmetric soliton (analytical) Symmetric soliton (numerical) Antisymmetric soliton (numerical)

0

0.2

0.4

γ /κ

0.6

0.8

1

Fig. 20 The analytically predicted stability border (102) for the PT -symmetric solitons and its counterpart produced by systematic simulations of the perturbed evolution of the solitons, as per Driben and Malomed (2011a). An effective numerical stability border for the PT -antisymmetric solitons is shown too, although all the antisymmetric solitons are, strictly speaking, unstable (Alexeeva et al. 2012) (the numerically identified stability area for them implies very weak instability). The solitons with amplitude  are stable at  < max . In this figure, = is identical to , as the inter-core coupling coefficient is fixed by scaling to be  D 1; see the text

2 |u|2

1.5 1 0.5 0 20

15

10 Z

5

0

–40

–20

0

20

40

t

Fig. 21 The elastic collision between stable PT -symmetric solitons with  D 0:7, boosted by frequency shift  D ˙5 at  D 0:7; see Eq. (103). The figure is shown as per Driben and Malomed (2011a)

impossible for them. Accordingly, the instability of solitons with  > max leads to a blowup of the pumped field, u, and decay of the attenuated one, v, in direct simulations (not shown here). The presence of the gain and loss terms in Eqs. (96) and (97) does not break their Galilean invariance, which suggests to consider collisions between moving stable solitons, setting them in motion by means of boosting, i.e., replacing fu; vg ! fu; vg exp.˙i t /

(103)

in the initial state ( z D 0), with arbitrary frequency shift . Simulations demonstrate that the collisions are always elastic; see a typical example in Fig. 21.

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B. A. Malomed

The limit case of  D 1 may be considered as one of “supersymmetry,” because the inter-core coupling constant and gain-loss coefficients are equal in Eqs. (96) and (97) in this case. According to Eq. (102), the stability region of the solitons in the supersymmetric systems shrinks to nil. The linearization of Eqs. (96) and (97) with  D 1 around the NLSE solution (98) leads to the following equations for perturbations ıu and ıv: O .ıu C i ıv/ D 0; L

(104)

O .ıu  i ıv/ D 2i  .ıu C i ıv/ ; L

(105)

 O where Lıu  i @z C .1=2/@t t C 2jU j2 ıu C U 2 ıu is the NLSE linearization operator. If the underlying NLSE solution is stable by itself, Eq. (104) produces no instability, while Eq. (105) gives rise to a resonance, as .ıu C i ıv/ is a zero O According to the linear-resonance theory (Landau and Lifshitz mode of operator L. 1988), the respective perturbation .ıu  i ıv/ is unstable, growing z (rather than exponentially). Direct simulations of Eqs. (96) and (97) with  D 1 confirm that the solitons are unstable, the character of the instability being consistent with its subexponential character (Driben and Malomed 2011a). The “supersymmetric” solitons may be stabilized by means of the management technique (Malomed 2006), which, in the present case, periodically reverses the common sign of the gain-loss and inter-core coupling coefficients, between  D K D C1 and 1 (recall K is the coefficient of the inter-core coupling, which was scaled to be 1 above, and may now jump between C1 and 1). Flipping  between C1 and 1 implies switch of the gain between the two cores, which is possible in the experiment. The coupling coefficient, , cannot flip by itself, but the signal in one core may pass -shifting plates, which is tantamount to the periodic sign reversal of K. Ansatz (98) still yields an exact solutions of Eqs. (96) and (97) with coefficients  D K subjected to the periodic management. On the other hand, the replacement of K by the periodically flipping coefficient destroys the resonance in Eq. (105). In simulations, this management mode indeed maintains robust supersymmetric solitons (Driben and Malomed 2011b).

Conclusion Dual-core optical fibers is a research area which gives rise to a great variety of topics for fundamental theoretical and experimental studies, as well as to a plenty of really existing and potentially possible applications to photonics, including both traditional optics and plasmonics. While currently employed devices based on dualcore waveguides operate in the linear regime (couplers, splitters, etc.), the use of the intrinsic nonlinearity offers many more options, chiefly related to the use of

10 A Variety of Dynamical Settings in Dual-Core Nonlinear Fibers

467

self-trapped robust modes in the form of solitons. In terms of fundamental studies, solitons in dual-core fibers are the subject of dominant interest. Theoretical studies of solitons in these systems had begun about three decades ago (Friberg et al. 1987, 1988; Heatley et al. 1988; Trillo et al. 1988; Trillo and Wabnitz 1988; Wright et al. 1989; Paré and Florja´nczyk 1990; Snyder et al. 1991; Maimistov 1991; Winful and Walton 1992; Walton and Winful 1993; Romagnoli et al. 1992; Akhmediev and Ankiewicz 1993a; Soto-Crespo and Akhmediev 1993; Chu et al. 1993; Ankiewicz et al. 1993; Akhmediev and Ankiewicz 1993b). The earlier works and more recent ones have produced a great advancement in this field, with the help of analytical and numerical methods alike. The present chapter is focused on reviewing the results obtained, chiefly, in three most essential directions: the spontaneous symmetry breaking of solitons in couplers with identical cores and the formation of asymmetric solitons; the creation of stable dissipative solitons in gain-carrying nonlinear fibers (actually, fiber lasers), stabilized by coupling the active (pumped) core to a parallel lossy one; and the stability of solitons in PT symmetric couplers, with equal strengths of gain and loss carried by the parallel cores. While the analysis of these areas has been essentially completed, taking into account both early and recent theoretical results, there remain many directions for the extension of the studies. In particular, a natural generalization of dual-core fibers is provided by multi-core arrays, which allow the creation of self-trapped modes which are discrete and continuous along the directions across the fiber array and along the fibers, respectively. These modes include semi-discrete solitons, which may be expected and used in many settings (Aceves et al. 1994a,b, 1995; Matsumoto et al. 1995; Blit and Malomed 2012). Another generalization implies the transition from 1D to 2D couplers, represented by dual-core planar optical waveguides. The consideration of spatiotemporal propagation in such a system makes it possible to predict the existence of novel species of 2D stable “light bullets” (spatiotemporal solitons (Silberberg 1990)), such as spatiotemporal vortices (Dror and Malomed 2011), solitons realizing the optical emulation of spin-orbit coupling (Kartashov et al. 2015; Sakaguchi and Malomed 2016), and 2D PT -symmetric solitons (Burlak and Malomed 2013; Sakaguchi and Malomed 2016). The most challenging problem is that, as yet, there are very few experimental results reported for solitons in dual-core and multi-core systems. One of experimental findings is the creation of semi-discrete “light bullets” (Minardi et al. 2010), including ones with embedded vorticity (Eilenberger et al. 2013) (actually, in a transient form), in three-dimensional arrays of fiber-like waveguides permanently written in bulk samples of silica. Further development of experimental studies in this vast area is a highly relevant objective. Acknowledgements I thank Professor Gang-Ding Peng for his invitation to join the production of this volume and to write the present chapter.

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S. Trillo, S. Wabnitz, Coupling instability and power-induced switching with 2-core dualpolarization fiber nonlinear coupler. J. Opt. Soc. Am. B 5, 483–491 (1988) S. Trillo, S. Wabnitz, E.M. Wright, G.I. Stegeman, Soliton switching in fiber nonlinear directional couplers. Opt. Lett. 13, 672–674 (1988) S. Trillo, G. Stegeman, E. Wright, S. Wabnitz, Parametric amplification and modulational instabilities in dispersive nonlinear directional couplers with relaxing nonlinearity. J. Opt. Soc. Am. B 6, 889–900 (1989) S.C. Tsang, K.S. Chiang, K.W. Chow, Soliton interaction in a two-core optical fiber. Opt. Commun. 229, 431–439 (2004) Y.J. Tsofe, B.A. Malomed, Quasisymmetric and asymmetric gap solitons in linearly coupled Bragg gratings with a phase shift. Phys. Rev. E 75, 056603 (2007) A.V. Ustinov, H. Kohlstedt, M. Cirillo, N.F. Pedersen, G. Hallmanns, G. Heiden, Coupled fluxon modes in stacked Nb/AlOx /Nb long Josephson junctions. Phys. Rev. B 48, 10614–10617 (1993) I.M. Uzunov, R. Muschall, M. Gölles, Y.S. Kivshar, B.A. Malomed, F. Lederer, Pulse switching in nonlinear fiber directional couplers. Phys. Rev. E 51, 2527–2537 (1995) M. van Hecke, Coherent and incoherent structures in systems described by the 1D CGLE: experiments and identification. Phys. D 174, 134–151 (2003) W. van Saarloos, P.C. Hohenberg, Pulses and fronts in the complex Ginzburg-Landau equation near a subcritical bifurcation. Phys. Rev. Lett. 64, 749–752 (1990) A. Villeneuve, C.C. Yang, P.C.J. Wigley, G.I. Stegeman, J.S. Aitchison, C.N. Ironside, Ultrafast alloptical switching in semiconductor nonlinear directional couplers at half the band-gap. Appl. Phys. Lett. 61, 147–149 (1992) Y.I. Voloshchenko, Y.N. Ryzhov, V.E. Sotin, Stationary waves in nonlinear, periodically modulated media with large group retardation. Zh. Tekh. Fiz. 51, 902–907 (1981) [Sov. Phys. Tech. Phys. 26, 541–544 (1982)] D.T. Walton, H.G. Winful, Passive mode locking with an active nonlinear directional coupler: positive group-velocity dispersion. Opt. Lett. 18, 720–722 (1993) H.G. Winful, D.T. Walton, Passive mode locking through nonlinear coupling in a dual-core fiber laser. Opt. Lett. 17, 1688–1690 (1992) E.M. Wright, G.I. Stegeman, S. Wabnitz, Solitary-wave decay and symmetry-breaking instabilities in two-mode fibers. Phys. Rev. A 40, 4455 (1989) Y.D. Wu, Coupled-soliton all-optical logic device with two parallel tapered waveguides. Fiber Integr. Opt. 23, 405–414 (2004) A. Zafrany, B.A. Malomed, I.M. Merhasin, Solitons in a linearly coupled system with separated dispersion and nonlinearity. Chaos 15, 037108 (2005)

Part III Optical Fiber Fabrication

Advanced Nano-engineered Glass-Based Optical Fibers for Photonics Applications

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M. C. Paul, S. Das, A. Dhar, D. Dutta, P. H. Reddy, M. Pal, and A. V. Kir’yanov

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Importance of the Nano-engineered Glass-Based Optical Fiber . . . . . . . . . . . . . . . . . . . . . . . . The Basic Material of Nano-engineered Glass-Based Optical Fiber . . . . . . . . . . . . . . . . . . Importance of Ceramic Oxides in Nano-engineered Glass-Based Optical Fiber . . . . . . . . Mechanism to Develop Nano-engineered Glass-Based Optical Fiber . . . . . . . . . . . . . . . . . . . Fiber Drawing Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabrication of Erbium-Doped Nano-engineered Zirconia-Yttria-Alumina-Phospho-Silica (ZYAPS) Glass-Based Optical Fiber . . . . . . . . . . . Material Characterization of Erbium-Doped Nano-engineered ZYAPS Glass-Based Optical Preform and Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Optical Performance of Erbium-Doped Nano-engineered ZYAPS Glass-Based Optical Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabrication of Erbium-Doped Nano-engineered Scandium-Phospho-Yttria-Alumina-Silica (SPYAS) Glass-Based Optical Fiber . . . . . . . . . . Material Characterization of Erbium-Doped Nano-engineered SPYAS Glass-Based Optical Preform and Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Optical Performance of Erbium-Doped Nano-engineered SPYAS Glass-Based Optical Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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M. C. Paul () · S. Das · A. Dhar · D. Dutta · M. Pal Fiber Optics and Photonics Division, CSIR-Central Glass and Ceramic Research Institute, Kolkata, India e-mail: [email protected]; [email protected]; [email protected]; [email protected] P. H. Reddy Academy of Scientific and Innovative Research (AcSIR), IR-CGCRI Campus, Kolkata, India e-mail: [email protected] A. V. Kir’yanov Centro de Investigaciones en Optica, Guanajuato, Mexico e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 G.-D. Peng (ed.), Handbook of Optical Fibers, https://doi.org/10.1007/978-981-10-7087-7_72

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Fabrication of Multielement (P-Yb-Zr-Ce-Al-Ca) Fiber for Moderate-Power Laser Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material Characterization of Multielement (P-Yb-Zr-Ce-Al-Ca) Optical Preform and Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Optical Performance of Multielement (P-Yb-Zr-Ce-Al-Ca) Optical Fiber . . . . . . . . . Fabrication of Chromium-Doped Nano-phase Separated Yttria-Alumina-Silica (YAS) Glass-Based Optical Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material Characterization of Chromium-Doped Nano-phase Separated YAS Glass-Based Optical Preform and Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Optical Performance of Chromium-Doped Nano-phase Separated YAS Glass-Based Optical Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Nano-engineered glass-based silica optical fibers doped with different rare earth ions and transition metal ions is a new research area with potential application towards fiber-based amplifier, laser, and sensors. This chapter describes the basic material, fabrication techniques, and related material as well as the optical properties of nano-engineered glass-based optical fibers doped with erbium/ytterbium, erbium/scandium, ytterbium, and chromium for photonics applications. Accordingly, we present the development of erbium (Er)-doped zirconia-yttria–alumina-phospho-silica glass-based optical fibers and their application towards multichannel amplification in the C-band region under core as well as cladding pumped configuration. A new class of Erdoped nano-engineered scandium-yttria-alumina-silica glass-based optical fibers with proper annealing is also discussed which exhibited a flat-gain spectrum suitable for broad-band optical amplification in the CC L band region having a maximum gain of 39.25 dB at 1560 nm. Application of materials towards a burning issue in case of high power Yb-doped laser called photodarkening has been reported where developed multielement (P-Yb-ZrCe-Al-Ca) nano-phase separated silica-glass-based optical fiber reveals that multielement-Yb-doped fiber is a promising candidate for laser applications with enhanced photodarkening resistivity. Lastly focused fabrication and properties of transition metal chromium-doped nano-phase separated yttria-alumina-silica glass-based optical fibers which may be suitable as saturable absorber with potential application towards Q-switching for 1–1.1-m all-fiber ytterbium lasers. Keywords

Nano-engineered glass · Rare-earth doped nano-particles · Optical fiber · Saturable absorber · Photodarkening effect · Fiber laser · Optical amplifier · Nonlinear refractive index · Broadband light source

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Introduction Nano structure of optical material plays an important role at current nano-science for manipulating and enhancing light-matter interactions for improving fundamental device properties. Now-a-days fiber laser, optical amplifier, and broadband light source are the most essential opto-electronic devices in modern communication systems. Recently an interesting research area being attempted across the globe to develop optical devices comprised of rare earth (RE) and different transition metal-doped nano-structured materials-based optical fibers for their use in telecommunications as broadband light source, optical amplifier, and high power fiber lasers with low photo-darkening phenomena. Accordingly, research on nanostructured optical fibers elaborated by incorporating dielectric metallic nanoparticles, silicon nanoparticles, semiconductor nanoparticles, phase-separated dielectric nonmetallic nanoparticles, or quantum dots in an amorphous matrix attracted much attention. These nanoparticles dispersed in a silica matrix-based optical fibers exhibit large nonlinear optical properties and offer a great potential for optical amplification and lasing application as the RE ions doping concentration can be higher than conventional amorphous medium. Moreover, the energy transfer process in these rare-earths doped nanoparticles containing optical fiber leads to exotic luminescence properties. The well-known silica-based erbium-doped fiber amplifiers (EDFA) can produce gain in the C (1530–1565 nm) (Mears et al. 1987) and L bands (1570–1605 nm) (Massicott et al. 1990). The EDF based on fluoride and tellurite fibers are also reported to be applicable in the C and L bands, while the thulium-doped fiber amplifiers (TDFA) exhibit gain in the S band (Kasamatsu et al. 1999). Furthermore, the praseodymium-doped fiber amplifiers can operate in the O band (1260– 1360 nm) (Ohishi et al. 1991) and holmium-doped ones can access the U band (1625–1675 nm) window and beyond. On the other hand, fiber Raman amplifiers can cover a broad spectral region, provided a suitable fiber and pump source to be used. However, neither of the above mentioned amplifiers can fully cover the 1100–1500 nm range with a single “active” fiber to be used. Bismuth- and chromium-doped fibers having pronounceable absorption and fluorescence bands within the 1100–1500 nm range are currently being investigated. Erbium (Er)-doped fiber amplifiers (EDFA) have become established, since the invention in 1987 (Mears et al. 1987), as a standard component in telecommunication networks, facilitating information exchange worldwide due to their high performance and cost effectiveness. Broad gain at important low-dispersion wavelengths covering the telecom C-band (1525–1565 nm) and L-band (1565– 1610 nm) (Atkins et al. 1989; Sun et al. 1997), low noise (Wysocki et al. 1997), low-loss and compatibility with fiber lightwave systems (Giles and Desurvire 1991; Bergano and Davidson 1996) make EDFA an excellent fit for signal amplification with high optical efficiency at various points of such networks. Furthermore, EDFA, especially in the eye-safe spectral region 1500–1700 nm, continue to be useful in a wide range of other applications, including optical communication

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(Zimmerman and Spiekmann 2004), range finding, remote sensing (Wysocki et al. 1994), ultra-high bit-rate telecom transmission systems (Zhou et al. 2010), and freespace communications (Ma et al. 2009). The importance of the development work becomes evident from expanding global optical amplifier market which reached $900 million in 2012 and is anticipated to reach $2.8 billion by 2019 (NEW YORK 2013). To achieve high power Er-doped amplifier, it is essential to dope Er-ions at high concentration level which requires use of different glass host other than pure silica to overcome clustering problem. Accordingly, over the past several years extensive researches have been carried out on different materials for incorporation of Er-ions, such as silica glass modified with incorporation of GeO2 , Al2 O3 and P2 O5 , telluride glass, phosphate glass chalcogenide, bismuthate, fluorozirconate, lithium niobate, lanthanum (Wang et al. 2003; Yamada et al. 1996; Jha et al. 2000; Naftaly et al. 2000; Jiang et al. 2000; Harun et al. 2010). Literature survey shows that different materials have different impacts on the performance of the developed waveguide or amplifier. Some materials allow higher Er concentrations to be realized without detrimental effects of concentration quenching (Snoeks et al. 1996) and cluster formation (Gill et al. 1996), thereby providing higher gain for a compact device. Although many nonsilica-based fibers doped with Er offer high quantum efficiencies for 1.5-m transition, their material properties may not be suitable for practical devices. Moreover, owing to the difference in melting temperatures, such fibers cannot be spliced with standard single-mode fibers using a standard splicing machine. On the contrary, silica-based fiber doped with erbium continues to be the most preferable choice due to its proven reliability and compatibility with conventional fiber-optic components (Cheng et al. 2009; Harun et al. 2009). Due to unavailability of high power single mode pump lasers, development of erbium-doped fiber (EDF) is a challenging task and researchers are employing a double-clad erbium co-doped with ytterbium fiber amplifier (EYDF) in which the signal light propagates in the core and pump light propagates in the first cladding around the core. Such a double-clad EYDF amplifier can use watt-class multimode laser diodes, with promise of realizing amplifier of high output power. Compared with EDF, the co-doping with Yb ions considerably improves the pump absorption. In EYDF, the ytterbium can absorb a pump photon within its spectral band of 800– 1100 nm and be excited to the 2 F5/2 state, from which it can transfer energy to 4 I11/2 state of an erbium ion. The ions in 4 I11/2 then make a nonradiative transition to 4 I13/2 , forming a population inversion between 4 I13/2 and 4 I15/2 , thereby producing amplification of incident optical signal around 1550 nm. An all-fiber format is of great interest outside the laboratory environment. Among single mode fiber sources at these wavelengths, the highest output power of 150 W with 33% efficiency was achieved using Er/Yb co-doped fibers cladding pumped at 915 or 976 nm (Jeong et al. 2005). However, for the maximum output power, amplified spontaneous emission (ASE) from Yb ions at 1 m reaches 50% of the laser power at 1560 nm, leading to a rollover of the slope efficiency and raising the difficult issue of getting rid of the parasitic (non-eye-safe) radiation. In the recent past, significant success has been obtained in power scaling of in-band pumped Er-doped fiber (EDF) lasers and amplifiers (Zhang et al. 2011; Jebali et al. 2012;

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Supradeepa et al. 2012; Lim et al. 2012). In such approach, the reduced quantum defect leads to a very attractive lasing slope efficiency of 75% (Jebali et al. 2012). However, the electrical to optical efficiency of the pump diode sources or Er–Yb fiber lasers at 1532 nm (Zhang et al. 2011; Jebali et al. 2012) and Raman fiber lasers at 1480 nm (Supradeepa et al. 2012) is at least two times lower than that of usual pump diodes at 976 nm, making the overall electrical to optical efficiency in the demonstrated in band pumped lasers lower than 15%. It appears that the double-clad (DC) Yb-free EDF pumped at 976 nm is the simplest architecture. Slope efficiencies of 24% (Kuhn et al. 2011a) in a single-mode fiber and 30% in a multimode fiber have been reported (Kuhn et al. 2011b). By decreasing the pump-cladding diameter and using the suitable core glass matrix, a slope efficiency of 32% (>7 W at 1575 nm with potential power scaling) in a single-mode EDF has been demonstrated (Kotov et al. 2012). Very recently L. V. Kotov et al. demonstrated 75 W 40% efficiency single-mode all-fiber erbium-doped laser cladding pumped at 976 nm (Kotov et al. 2013). Thus, it is essential to carry out basic research in order to identify a suitable glass host for fabrication of EDF for higher optical amplification efficiency. Ytterbium (Yb)-doped fiber lasers due to their many folds advantages, namely, simple energy level diagram, small quantum defect, long lifetime at the excited state (1 ms), broad absorption (800–1080 nm), and emission (970–1200 nm) bands (Zervas and Codemard 2014) over the conventional bulk state solid state lasers, find variety of applications covering materials processing (cutting, grinding, and engraving) medical and military applications (Jauregui et al. 2013) using continuous wave (CW) power level as high as 100 kW have been reported (Fomin et al. (2014); Gapontsev et al. 2014). Nevertheless, research is continuing to achieve higher power for specific industrial and defense applications. This can be achieved by enhancement of Yb concentration with improved waveguide design and optimization of laser cavity design to minimize different nonlinear effects, e.g., stimulated brillouin scattering (SBS), stimulated Raman scattering (SRS) besides photodarkening (PD) phenomenon. Photodarkening has been a hot topic of research interest since 2005 after Koponen et al. (2005) first reported about photodarkening in details, although prior to that Paschota et al. reported about unsaturable loss in Yb-doped fiber (Paschotta et al. 1997). The photodarkening effect can be defined as a time dependent increase of background absorption in a RE-doped fiber’s core which supposedly originated from a multistep multiphoton absorption process, with the main threats to the efficiency of the Yb-doped gain media leading to degradation of laser performance and long-term stability. Photodarkened Yb-doped fibers demonstrate absorption centered in UV and extends to the VIS region with its tail stretching towards NIR region, the latter leads to degradation of the fibers lasing property (Mattsson 2011). However, PD is not a solely Yb-specific problem, but is as well observed in other RE ions like Tm, Tb, Ce, Pr doped into silica fiber (Broer et al. 1993; Atkins and Carter 1994; Behrens and Powell 1990). Accordingly, it is important to control and diminish PD in order to achieve enhanced power level and to improve the performance and reliability of Yb-doped high-power fiber lasers. Although the actual mechanism behind the origin of PD in Yb-doped silica is still an open subject of research, in general formation of defect centers, also

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known as “color center,” seems to be the prime reasons for PD. Different theories like formation of oxygen deficiency centers (ODC) (Yoo et al. 2007), Yb2C related charge-transfer-states (CTS) were reported by different groups (Engholm and Norin 2008; Guzman Chavez et al. 2007). On the other hand, different processes related to PD mitigation (completely or partially) were also reported, namely, oxygen or hydrogen (Jaspara et al. 2006) loading, temperature annealing (Soderlund et al. 2009), exposure to UV or VIS light (Guzman Chavez et al. 2007; Manek-Honninger et al. 2007) using pump conditions (Jetschke et al. 2007), etc. However, none of the abovementioned methods can be considered as a practical solution as all of them directly impact on fiber coating materials; so research is still pursuing to find out an acceptable route to mitigate PD. It has been observed that higher Al-doping level (Kitabayashi et al. 2006), lower Yb-ion concentration (Morasse et al. 2007), co-doping with phosphorous (P) (Jetschke et al. 2008; Sahu et al. 2008) and cerium (Ce) (Engholm et al. 2009; Malmstrom et al. 2010) can increase PD resistivity. Higher Al doping level increases numerical aperture (NA) and degrades beam quality and thus is not a worth solution. Use of lower Yb concentration unavoidably leads to longer fiber to be used for lasing, which strengthens unwanted nonlinear effects and ultimately prevents high-power operation. In this context, co-doping of Yb-doped fiber with P, though enabling superior PD resistivity, is associated with increased base loss, lower emission, and absorption cross sections besides a central dip formation during fabrication, leading to degradation of lasing performance. Furthermore, Ce-doping could be really an attractive option to reduce PD, but it increases NA and has constraint for practical applications. Very recently, a glass composition with equal Al and P ratio was shown (Unger et al. 2009) to be a promising alternative to enhance PD resistivity, but this composition generally produces triangular refractive index (RI) profile and inhomogeneity along the length. Considering all the above referred information, searching for a relevant solution is going on to alter core glass composition for minimizing formation of PD-related color center even at a higher Yb ion concentration. Fiber-based amplifiers which cover the 1100–1500 nm range with a single “active” fiber are still under development as both Bi- and Cr-doped fiber fabrication which can operate in this region is challenging. Nevertheless, between Bi and Cr, Cr-doped fiber fabrication is the most challenging task. Among the transition metal (TM) doped fibers, Cr4C :YAG doped fiber having broadband fluorescence that covers 1100–1600 nm range (Lo et al. 2005; Huang et al. 2006, 2013; Lai et al. 2011) is of great importance for implementing broadly tunable solid-state lasers for 1400–1500 nm spectral range as well as a reliable saturable absorber for ytterbium- and neodymium-doped lasers at 1000–1100 nm range (Angert et al. 1988; Eilers et al. 1993; Borodin et al. 1990; Il’ichev et al. 1998a, b; Mellish et al. 1998; Dussardier et al. 2011). On the other hand, Cr4C :YAG doped fibers can fit a variety of applications (Alcock 2013), including injection-seeded optical parametric oscillators (OPOs), spectroscopy, optical coherence tomography (OCT), “eye-safe” ranging, optical time domain reflectometry, tissue welding as well as pumping EDFA and erbium doped fiber lasers. Accordingly, Cr4C as

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active ion and yttrium-aluminum garnet (YAG) as a host is a potential candidate which needs further research. Embedding chromium into a glassy (silicate) matrix directly was not so successful to meet the practical applications highlighted above due to the fact that chromium exhibits different oxidation states in silicates while there are definitive technological difficulties in attempt to stabilize a wanted state (especially Cr4C ions). Each oxidation state has its own potential applications: e.g., Cr4C ions can be used as an “active” medium for lasing and amplifying around 1400 nm (Zhavoronkov et al. 1995; Dong et al. 2001) while Cr3C /Cr4C codoping – for realizing tunable solid-state lasers (Nikolov et al. 2004; Sennaroglu 2002) usefulness of the Cr4C species as saturable absorbers (Dussardier et al. 2011; Kück 2001). However, the different oxidation states arise due to change in glass compositions, processing temperature, and upon an employed preparation technology. Several attempts have been made to fabricate chromium-doped silica fibers by different authors worldwide. For instance, earlier workers (Lo et al. 2005; Huang et al. 2006, 2013; Lai et al. 2011) reported the laser-heated pedestal growth (LHPG) method to fabricate a single-crystal fiber, but the process implementation was intricate due to the technological complexity associated with single-crystal fiber drawing. Another method known as the rod-in-tube (RIT) technique was exploited which results in better glass uniformity with small core dimension (Huang et al. 2007, 2010). Nevertheless, increasing chromium concentration in Cr:YAG rod in the RIT method was difficult, and thus powder-in-tube (PIT) method was introduced to increase chromium ions concentration further (Huang et al. 2011). Many researchers (Felice et al. 2000; Dvoyrin et al. 2003; Glazkov et al. 2002; Abramov et al. 2014) fabricated silica-glass-based Cr4C -doped fibers based on alumino-silica, germanosilica, and gallium-silica glasses and optical fibers, obtained using the well-known Modified Chemical Vapor Deposition (MCVD) coupled with solution doping (SD) technique, though they were not able to demonstrate broadband emission of Cr4C ions. The only successful approaches to get near-IR (1 m) fluorescence stemming from Cr4C ions presence were proposed in Zhuang et al. (2009), and Wang and Shen (2012), but in that case the network permitting stabilization of Cr4C species was glass ceramics. Thus, fabrication of Cr-doped fiber with a suitable glass host to stabilize Cr4C ion through conventional modified chemical vapor deposition (MCVD) coupled with solution doping process is a challenging task. In this book chapter, we have thus focused to develop suitable glass host materials comprised of nano-engineered phase separated silica-based optical fiber, fabricated through conventional MCVD-SD technique to tackle the abovementioned problems. The principles associated with fabrication techniques along with material and optical characterization as well as fiber performance will be discussed in details.

Importance of the Nano-engineered Glass-Based Optical Fiber Since pure silica is not a suitable matrix for rare earth (RE) ions due to low solubility of RE ( few hundreds of ppm) higher RE concentration leads to phase separation and reduction in fiber performance. If the phase of micron-size aggregates is

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separated from a homogeneous glass matrix, it causes a fatal increase of spectral attenuation of the fibers and must be prevented to develop low loss RE-doped fibers. Modification of glass matrices mainly with Al2 O3 , P2 O5 or GeO2 has been practiced in silica fibers for years with the aim to increase the solubility of REs in the fiber core as well as to enhance the emission property. However, such co-dopants are not sufficient to achieve high concentration RE-doping without clustering phenomena for making of specialty fibers under high power applications. Successful insulation of RE-ions from matrix vibrations is required by appropriate ion-site engineering through the formation of phase-separated nano-particles. This approach is proposed via encapsulation of dopants inside glassy or crystalline nanoparticles embedded in the fiber glass under suitable doping host composition using ceramic oxides, such as Al2 O3 , Sc2 O3 , ZrO2 , and Y2 O3 followed by post thermal annealing process. In such nano-engineering glass-based optical fibers, the basic material silica serves as a support for providing optical and mechanical properties to the fiber, whereas the spectroscopic properties would be controlled by the composition along with nature of nano-particles. The thermal shock resistance of the nano-engineering host of REs will be increased under high power applications. The sizes of nano-particles should be kept within 5–10 nm ranges to reduce the scattering loss as well as increases the optical transparency of the glass.

The Basic Material of Nano-engineered Glass-Based Optical Fiber The nano-engineering glass is based on multielements doped into silica glass matrix containing ceramic oxides, such as Al2 O3 , ZrO2 , Y2 O3 along with RE such as Er, Yb, Tm, Ce and transition metals such as Cr, Sc. In some cases, minor amount of P2 O5 is used for making of fiber laser and optical amplifier where P2 O5 acts as nucleating agent to accelerate the growth of formation of phase separated particles upon heating through thermal perturbation owing to the higher field strength difference (>0.31) between Si4C and P5C . Accordingly, in this chapter, the fabrication of the following nano-engineering glass-based optical fibers have been discussed. • Er-doped nano-engineered scandium-yttria-alumina-silica (SYAS) glass-based optical fiber for brand optical amplification with flat gain • Erbium-doped zirconia-yttria-alumina-phospho-silica glass-based optical fiber for high power optical amplification • Er-doped nano-engineered scandium-yttria-alumina-silica (SYAS) glass-based optical fiber for broadband optical communication system • Fabrication of multielement (ME) (P-Yb-Zr-Ce-Al-Ca) nano-phase separated silica glass-based core region of optical fiber using conventional modified chemical

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vapor deposition (MCVD) process coupled with solution doping technique and study of their photodarkening resistant behavior • Fabrication of yttria-alumina-silica nano-engineering glass host-based chromiumdoped phase-separated particles containing optical fibers using the MCVD process in conjunction with the SD technique along with study of their spectroscopic properties

Importance of Ceramic Oxides in Nano-engineered Glass-Based Optical Fiber The use of different ceramic oxides has multiple advantages as mentioned below. • It helps to reduce the clustering phenomena of RE ions significantly by enhancing solubility of RE compared to pure silica glass matrix. • Increases the refractive index of silica glass due to the larger sizes of Al, Zr, and Y cations as well as their enhanced ionic polarizability. • Increases the optical nonlinearity of the doping host of REs because of the formation of large number of nonbridging oxygen (NBO) and formation of phase-separated nano-particles. Nonbridging oxygens have high ionicity and are easily distorted by applied optical-electric field as the defects generated from the phase-separated nano-particles. • Zirconia (ZrO2 ) doping into silica glass increases its physical and chemical properties, including hardness, wear resistance, low coefficient of friction, elastic modulus, chemical inertness, ionic conductivity, and electrical properties. • Y2 O3 serves as an attractive host material for laser application since it is a refractory oxide with a melting point of 2380 ı C a very high thermal conductivity, kY2O3 D 27 W/mK, two times YAG’s one, kYAG D 13 W/mK. Another interesting property allowing radiative transitions between electronic levels is the dominant phonon energy of 380 cm1 which is one of the smallest phonon cut-offs among oxides. • Scandium oxide is a cubic crystal belonging to the Ia3 space group (space group number 206) having the lattice constant of 9.8459 ı A while Er3C :Sc2 O3 sesquioxide is a promising candidate for high energy, eye-safe solid-state lasers as it can operate with a very low quantum defect and high thermal conductivity Peters et al. (2002; Ter-Gabrielyan et al. (2008)).

Mechanism to Develop Nano-engineered Glass-Based Optical Fiber The following diagram shows the general steps involved towards development of nano- engineering glass-based optical fibers:

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M. C. Paul et al. Deposition of porous soot layer of optimized vapour phase composition Glass modifiers incorporated by solution doping technique Incorporation occur through viscous sintering phenomena At sintering temperature the core glass will be in a metastable immiscibility under condition of TC < Tm where phase separation kinetics are faster than crystallization kinetics More and more negative value of the free energy change of the system for mixing of the oxide components the phase separation will be faster The fabricated preform is annealed under optimized condition based on glass composition prior to fiber drawing in order to achieve phase-separation with smaller particle size to reduce loss characteristics of final fiber

Fiber Drawing Process Fiber drawing from annealed preform was done using fiber drawing tower as shown in Fig. 1, with on-line dual resin coating. Depending upon the diameter and fiber drawing speed, the furnace temperature is set. At the same time, fiber is drawn down by a capstan puller at certain speed. After cooling down, the bare fiber is coated with two layers of UV–curable acrylate resin. High-quality coating with good coating concentricity and accurate dimensions is essential for getting high quality fibers. The purpose of the coating on the optical fiber is to protect the fiber from contact and foreign particles as these reduce the life of the fiber in terms of strength and static fatigue. Besides strength preservation, the coating must protect from microbending by being concentric and bubble free around the fiber and by having a stable performance in different environments. Out of two layers of UV-curable acrylate, a soft inner layer protects the fiber and a hard outer layer ensures good mechanical properties. The movement of the coating liquid occurs surrounding the surface of the fiber where fine bubbles can be trapped at the interface of fiber surface and hard coating. This can be avoided by controlling pressure and keeping the viscosity of the liquid constant during the drawing process before it enters the UV curing oven. Finally the fibers are wound on a precision spooling machine.

Fabrication of Erbium-Doped Nano-engineered Zirconia-Yttria-Alumina-Phospho-Silica (ZYAPS) Glass-Based Optical Fiber Erbium-doped optical fibers made based on ZYAPS glass employing modified chemical vapor deposition (MCVD) process in combination with the solution doping technique. Figure 2a shows the deposition of porous silica soot layer within the pure silica substrate tube of dimension 20 mm with thickness of 1.5 mm.

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Fig. 1 Photograph showing the view of the optical fiber drawing tower with picture of the neck down of preform coming out from the heated furnce before drawing the fiber

Figure 2b shows the actual collapsing process in MCVD set-up. The glass formers incorporated by the MCVD process are SiO2 and P2 O5 along with the glass modifiers Al2 O3 , ZrO2 , Er2 O3 , and Y2 O3 , which are incorporated by the solution doping technique using an alcohol-water mixture of suitable strength (1:5) to form the complex molecules ErCl3 .6H2 O, AlCl3 .6H2 O, YCl3 .6H2 O, and ZrOCl2 8H2 O. Small amounts of Y2 O3 and P2 O5 are added during the fabrication stage, which serve as a nucleating agent to increase the phase separation for generating Er2 O3 -doped microcrystallites in the core matrix of the optical preform (Paul et al. 2010, 2011; Pal et al. 2011). The inclusion of the Y2 O3 particulates into the host matrix serves the additional purpose of slowing down or eliminating changes in the ZrO2 crystal structure which exists in three distinct crystalline structures in a bulk glass matrix depending on the fabrication temperature. Another critical stage is annealing of the fabricated preform which was performed at 1100 ı C under optimized conditions in a closed furnace under heating and cooling rates of 20 ı C/min to generate Er2 O3 -doped ZrO2 rich nano-crystalline particles and subsequently fibers were drawn with on-line resin coating using fiber drawing tower. Details of representative fibers are presented in Table 1.

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Fig. 2 (a) Deposition of porous phospho-silica layer inside the silica substrate tube; (b) Collapsing is in progress for making Zr-EDF. (Reprinted with permission from reference 53, M. C. Paul, M. Pal, S. Das, A. Dhar and S. K. Bhadra; Journal of Optics 45, 260 (2016), Copyright @ The Optical Society of India, 2016, publisher Springer India)

Table 1 Nano-crystalline zirconia yttria alumina silica particles doped optical fibers Fiber No

Composition of doping host

NA

MEr-1

SiO2 -Al2 O3 -Y2 O3 -ZrO2 P2 O5 -Er2 O3 SiO2 -Al2 O3 -Y2 O3 -ZrO2 P2 O5 -Er2 O3 SiO2 -Al2 O3 -Y2 O3 -ZrO2 P2 O5 -Er2 O3

MEr-2 MEr-3

0.17

Core diameter (m) 10.5

Er ion concentration (ppm wt) 2800

0.19

10.0

3888

0.21

10.3

4320

Material Characterization of Erbium-Doped Nano-engineered ZYAPS Glass-Based Optical Preform and Fiber During fiber drawing at around 2000 ı C, the nano-crystalline host of ZrO2 was preserved in the silica glass matrix as confirmed by the TEM analyses with EDX spectra and electron diffraction patterns as shown in Fig. 3. Therefore, it could be ascertained that during fiber drawing at the high temperature further vitrification could be avoided. This clearly confirms the existence of ZrO2 crystallites within the host matrix. The average particle sizes are in the range 5–8 nm. The core and cladding geometry of the fiber was inspected by high resolution optical microscope (Olympus BX51). The core was homogeneous and had no observable defects at the interface between the core and the silica cladding. The average dopant percentages of the fiber samples were measured by electron probe microanalyses (EPMA) as shown in Table 2. Three different types of nano-engineered glass-based Zr-EDFs were fabricated with a variation of the doping levels of different co-dopants. The cross-sectional view of one of the fabricated Zr-EDF (MEr-2) is shown in Fig. 4a. It can be seen that it does not contain any core-clad imperfection which may affect bending loss

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Fig. 3 (a) TEM image of fiber core glass shows nano-crystallites (black spots) (b) electron diffraction pattern of nanoparticle of Zr-EDF (MEr-2) and (c) EDX analyses showing presence of Er Y, Al and Zr. (Reprinted with permission from reference 53, M. C. Paul, M. Pal, S. Das, A. Dhar and S. K. Bhadra; Journal of Optics 45, 260 (2016), Copyright @ The Optical Society of India, 2016, publisher Springer India) Table 2 Doping levels within core region of the fibers

ID MEr-1 MEr-2 MEr-3

Al2 O3 (mol%) 0.25 0.26 0.24

ZrO2 (mol%) 0.65 1.47 2.10

Er2 O3 (mol%) 0.155 0.195 0.225

of the fiber. The refractive-index profile of fiber MEr-1 is shown in Fig. 4b. The spectral attenuation curve of fiber MEr-2 is shown in Fig. 4c. The Zr-EDFs show small variation of the fluorescence lifetime at different pump powers varying from 50 to 300 mW.

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a 126.1

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Fig. 4 (a) Cross-sectional view (MEr-2), (b) refractive index profile (MEr-1) and (c) Loss curve of EDF (MEr-2). (Reprinted with permission from reference 53, M. C. Paul, M. Pal, S. Das, A. Dhar and S. K. Bhadra; Journal of Optics 45, 260 (2016), Copyright @ The Optical Society of India, 2016, publisher Springer India)

The fluorescence lifetimes of drawn fibers are determined and found to be around 8 and 10 ms under 100 mw pump power at 976 nm wavelength. MEr-1 shows a slightly lower fluorescence lifetime of around 8.0 ms as the fiber contains lowest doping level of ZrO2 among the three fibers under study. With a combination of both Zr and Al along with other co-dopants, we achieved high Er-concentration of 4320 ppm in the fiber MEr-3 without any phase separations and concentration quenching effect of REs. One of the conventional methods to test for quenching is the measurement of fluorescence lifetime of Er-ions at different pump powers. There must be concentration quenching of Er-ions if the fluorescence lifetime differs too much with different pump power (Keiichi Aiso et al. 2001). In order to check this property, we have measured the fluorescence lifetime of two different doped EDF hosts at different pump powers as shown in Fig. 5. The curves show that there was not much noticeable change of measured lifetime of Zr-EDF (MEr-3) that only changes from 10.65 to 10.70 ms with increasing pump power from 25 to 250 mW as in Fig. 5. This confirms that the lifetime of alumina-silica glassbased EDF normally varies from 10.25 to 10.58 ms. This phenomenon indicates that nano-crystalline ZrO2 prevents the formation of Er–Er bonds which reduces the quenching process of Er ions even at high doping levels of around 4320 ppm. The success is related to co-doping of ZrO2 with Al2 O3 . Both aluminum and zirconium

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Fig. 5 Fluorescence lifetime of three EDFs. (a) MEr-3 and (b) alumina-silica glass-based EDF at different pump powers. (Reprinted with permission from reference 53, M. C. Paul, M. Pal, S. Das, A. Dhar and S. K. Bhadra; Journal of Optics 45, 260 (2016), Copyright @ The Optical Society of India, 2016, publisher Springer India)

491

10.5 b 10.0

9.5

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250

ions surround the Er-ions and form a solvation shell thereby adjusting the charge balance and improving the solubility of Er ions into the host. This inhibits clustering of the Er-ions.

The Optical Performance of Erbium-Doped Nano-engineered ZYAPS Glass-Based Optical Fiber Four-channel amplification at different input signal levels 20, 25, and 30 dBm/ch at different pump powers (40, 50, and 60 mW) using one of the Zr-EDFs (MEr-3) published by our group (Pal et al. 2011) has been also evaluated. The amplified signals and output signal to noise ratio (SNR) were measured for three different pump powers of 40, 50, and 60 mW at 20 dBm/ch input signal level. Signal amplifications were increased in each channel and observed more rapid in the shorter wavelength side of C-band as pump power was increased, and the output SNRs are always >31 dB for each channel. The nature of signal amplifications and SNRs were measured for three different input signal levels (20, 25, and 30 dBm/ch) for constant pump power of 50 mW (Pal et al. 2011). It is observed that there is appreciable optical gain of at least 20 dB for 20 dBm/ch and a minimum output SNR of 21.8 dB for 30 dBm/ch input signal level. Both the gain and output SNRs would increase if pump power was further increased. The measured noise figures were within 4.2 to 4.7 dB. The critical passive losses of erbium-doped optical fiber in EDFA system arise from the OH – absorption that is present in doped glass which gives rise to strong absorption at 1380 nm wavelength. The presence of OH degrades the gain performance of the fibers due to the energy transfer from 4 I13/2 level to OH ions. The OH absorption losses of all the three fibers at 1380 nm wavelength are kept within 200–250 dB/km. On the other hand, splicing losses with standard SMF-28 fiber induces further loss in EDFA system which also reduces the pump efficiency of the fiber. All the splicing losses of all the three fibers are observed to be 0.04–0.05 dB.

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Fig. 6 Four-channel amplification in MEr-3 fiber with input signal level: 30 dBm/ch. (Reprinted with permission from reference 53, M. C. Paul, M. Pal, S. Das, A. Dhar and S. K. Bhadra; Journal of Optics 45, 260 (2016), Copyright @ The Optical Society of India, 2016, publisher Springer India)

The experimental results revealed that the Zr-EDF (MEr-3) is quite suitable for multichannel small-signal amplification. In our experiment, we achieved the minimum gain of 22.5 ˙ 0.5 dB for the input signal level of 30 dBm/ch where about 1.5 m length of the fiber was used in the experiment. The measured results of output power and SNR are shown in Fig. 6. We obtained gain difference (maxto-min gain) of less than 2 dB in MEr-3 sample. The maximum noise level was 28.5 dBm/nm for MEr-3. Moreover, MEr-3 showed an output SNR value of >22 dB. The motivation for choosing nanocrystalline zirconia dispersed in the silica glass matrix is to get low noise as well as better flatness of the gain spectrum at high doping level of Er ions. A major factor that influences the quenching process in REdoped materials is multiphonon relaxation. Zirconium oxide possesses a stretching vibration at about 470 cm1 , which is very low compared with that of Al2 O3 (870 cm1 ) and SiO2 (1100 cm1 ) (Patra et al. 2002). Hence, much attention is given to select such host materials with low phonon energy in order to improve the erbium ion solubility with reduced concentration quenching process and enhance the gain flatness and output SNRs for multichannel small-signal amplification. As stated earlier the composition of the doping host should be engineered properly for reducing the clustering effect of Er ions. We have tried to develop new compositional glass host for doping of Er ions with high ZrO2 content with the large cladding pump region. The microscopic image of hexagonal shaped low RI coated Er-doped fiber HPE-2 core and cladding spectral absorption curves are given in Fig. 7a, b, c. The core absorption is around 80 dB/m at 980 nm wavelength, whereas cladding absorption is around 2.0 dB/m at 980 nm. The different dopants are distributed uniformly along the diameter of the fiber, which is confirmed by electron probe microanalysis (EPMA) shown in Fig. 8a. A preliminary result of optical amplification

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122.23 μm

10.05 μm 128.51 μm

c

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1000 1200 1400 1600 Wavelength(nm)

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Fig. 7 (a) Cross-sectional view, (b) core absorption loss, and (c) cladding absorption loss of high concentration of erbium-doped fiber (HPE-2). (Reprinted with permission from reference 53, M. C. Paul, M. Pal, S. Das, A. Dhar and S. K. Bhadra; Journal of Optics 45, 260 (2016), Copyright @ The Optical Society of India, 2016, publisher Springer India)

b 10

Wt %

6

Al203

4

Y203

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Er203

0 –2 0

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10

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Ba0

0.0 –10

Zr02

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Main scale

a

–20 –30 –40 –50 –60 –70 1500 1540 1580 1620 1660 1700 1740 1780 Wavelength(nm)

Fig. 8 (a) Distribution of different dopants across the core diameter and (b) amplification curve of EDF (HPE-2) under 4 W pump power at 980 nm having I/P Signal C6 to 9 dBm with O/P signal of 500 mW. (Reprinted with permission from reference 53, M. C. Paul, M. Pal, S. Das, A. Dhar and S. K. Bhadra; Journal of Optics 45, 260 (2016), Copyright @ The Optical Society of India, 2016, publisher Springer India)

using a 4 m length of fiber under 4 W pump power having I/P signal C6 to 9 dBm is shown in Fig. 8b. The wideband ASE was observed for the yttria stabilized zirconiaalumina-phospho-silica glass host with hexagonal shaped cladding structure coated with low RI resin under different cladding pump power is given in Fig. 9.

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Fig. 9 ASE output for 6 m HPEr-5 for different pump powers for the wavelength range 1450–1650 nm. (Reprinted with permission from reference 53, M. C. Paul, M. Pal, S. Das, A. Dhar and S. K. Bhadra; Journal of Optics 45, 260 (2016), Copyright @ The Optical Society of India, 2016, publisher Springer India)

A wide band emission from 1530 to 1620 nm is observed. As the zirconia-rich Er2 O3 -doped phase-separated particles with partially crystalline character possesses lower phonon energy which gives rise to increase of larger emission cross section (Pal et al. 2011). The introduction of Zr4C allows one to avoid formation of Er3C clusters in silica host and consequently to make more intense luminescence (Paul et al. 2011). Replacing the intermediate Al2 O3 by a modifier ZrO2 , the number of nonbridging oxygen is expected to increase which makes the silica network structure more open. As a result, Zr-EDF has wider emission spectra as compared to silicaEDF, especially at the longer wavelengths at 1620 nm because of its larger emission cross-section (Paul et al. 2010, 2011). The widening of the ASE spectra towards a longer wavelength is believed to be a result of the Stark level of the Er3C ions in the Zr-EDF, which is separated to a larger degree due to intense ligand field. This is due to inhomogeneous energy level degeneracy that the ligand field of the zirconia host glass induced as a result of site-to-site variations, also known as the Stark effect, causing the widened optical transitions.

Fabrication of Erbium-Doped Nano-engineered Scandium-Phospho-Yttria-Alumina-Silica (SPYAS) Glass-Based Optical Fiber Development of solid-state Er:Sc2 O3 laser started recently although relevant spectroscopic investigations are still relatively rare due to limited availability of high optical quality material Fechner et al. (2008). Er-doped scandium-phospho-yttriaalumina-silica (SPYAS) glass-based fiber was fabricated using Modified Chemical Vapor Deposition (MCVD) process (Nagel et al. 1982) coupled with solution doping technique, by passing SiCl4 and POCl3 vapors through a slowly rotating high purity silica glass tube of outer diameter of 20 mm and inner diameter of 17 mm. An external flame source moved along the length of the tube as it

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rotated to heat it up under an optimized temperature of around 1550 ˙ 10 ı C, monitored by an IR pyrometer, synchronously moved with the oxy-hydrogen burner. This high temperature caused chloride vapors to oxidize, resulting in deposition of porous phospho-silica layer along the inner diameter of the tube. The glass formers, SiO2 and P2 O5 , were incorporated in the core matrix of the optical fiber through the MCVD process, with small amount of P2 O5 serving as a nucleating agent, to increase the phase separation along with the generation of Er-enriched microcrystallites. Then, the glass modifiers, Al2 O3 , Er2 O3 , Sc2 O3 , and Y2 O3 , were added through the solution doping technique using an alcohol/water mixture of suitable strength (1:5) to form the complex molecules ErCl3 .6H2 O, AlCl3 .6H2 O, YCl3 .6H2 O, and ScCl3 .6H2 O, diffusing into the silica glass matrix of the tube. Note that incorporating a small amount of Y2 O3 into the host matrix served to slow down or stop the changes in the crystalline structure formed and thereby preserving the mechanical strength and integrity of the final fiber. The fabricated preform was then cut into two portions. First portion of the preform was drawn into a fiber strand with diameter 125.0 ˙ 0.5 m through a conventional fiber drawing tower. The second portion of the preform was thermally annealed at 1200 ı C for 3 h with heating and cooling rate of 15 ı C/min in a closed furnace that maintained an optimized heating cycle and then was drawn to fiber. Note that the thermal annealing was a terminating step in the formation of the nano-engineered scandium-phospho-yttrium-alumina-silica (SPYAS) glassbased optical fiber. Because of the phase separation phenomenon, slight brownish coloration was observed in the core glass of the fiber. Once the fiber has been drawn, the primary and secondary coatings were added to the fiber; their uniformity was assured by controlling the flow pressure of the inlet gases into the coating resin vessels during the drawing process, as well as the proper alignment of the coating cup units. To ensure that a high-quality fiber was produced, its diameter was controlled throughout the drawing process (Bhadra and Ghatak 2013).

Material Characterization of Erbium-Doped Nano-engineered SPYAS Glass-Based Optical Preform and Fiber The optical microscopic views, TEM bright-field images, and electron diffraction (ED) patterns of Er-doped nano-engineered SPYAS glass-based fibers drawn from both the pristine and the annealed performs along with Sc-free EDF are demonstrated in Figs. 10 and 11, respectively. The microscopic views of the fibers cross-sections were measured by a highresolution optical microscope (Olympus BX51), connected to a high-resolution digital color camera. As seen from Figs. 10 and 11a, both the fibers have core of 12.5 m and are homogeneous without any core-clad imperfections, which otherwise might lead to a certain loss at bending. As also seen from Fig. 11a, the color of core glass is slightly brownish yellow in the fiber drawn from the annealed preform, being a result of the formation of glass-ceramic nature of the doping host. The morphology of the fiber core glass was studied by means of

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Fig. 10 Microscopic views of the pristine SPAYS (a) and Sc-free standard (b) Er-doped fibers; ED pattern (c) and TEM picture (d) of the SPAYS fiber. (Reprinted with permission from reference 75, P. H. Reddy, S. Das, D. Dutta, A. Dhar, A. V. Kir’yanov, M. Pal, S. K. Bhadra and M. C. Paul, Physica Status Solidi A 1700615 (2018), copyright @ 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

high-resolution transmission electron microscopy (TEM) (Tecnai G2 30ST, FEI Company, USA), producing the bright-field images shown in Figs. 10 and 11b. The nature of nanoparticles was evaluated from the ED patterns. For the fiber drawn from the pristine preform, the formation of nano-phase separated amorphous nanoparticles is presented in Figs. 10b and 11c. In turn, for the fiber drawn from the annealed preform, crystalline nature in core-glass is evidenced by Fig. 11b, c. Furthermore, the TEM along with ED pattern suggest the formation of crystalline nanoparticles of size 8–12 nm. Average doping distributions in the core region of the Er-doped SPYAS glassbased optical fiber were measured by Electron Probe Micro-Analyzer (EPMA), as shown in Fig. 12. From the distribution profiles shown, the following estimates of doping levels have been made: 1.256 wt% of Sc2 O3 , 5.85 wt% of Al2 O3 , 0.567 wt% of Y2 O3 , and 0.285 wt% of Er2 O3 . The Refractive Index (RI) profile of the Er-doped SPYAS glass-based optical fiber along with Sc free EDF, shown in Fig. 13, was measured using a fiber analyzer

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Fig. 11 (a) Microscopic view, (b) TEM picture, (c) high-resolution TEM, and (d) ED pattern of Er-doped nano-engineered SPYAS glass-based optical fibers drawn from the annealed perform. (Reprinted with permission from reference 75, P. H. Reddy, S. Das, D. Dutta, A. Dhar, A. V. Kir’yanov, M. Pal, S. K. Bhadra and M. C. Paul, Physica Status Solidi A 1700615 (2018), copyright @ 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

6

Dopant level (Wt%)

Fig. 12 Elemental analysis of the Er-doped SPYAS glass-based optical fiber. (Reprinted with permission from reference 75, P. H. Reddy, S. Das, D. Dutta, A. Dhar, A. V. Kir’yanov, M. Pal, S. K. Bhadra and M. C. Paul, Physica Status Solidi A 1700615 (2018), copyright @ 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

5 Al2O3 Sc2O3 Y2O3 Er2O3

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0.020 0.015 0.010 0.005

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Position (Micron) Fig. 13 RI Profile of the Er-doped nano-engineered SPYAS glass-based (a) and Sc-free standard Er-doped fiber (b). (Reprinted with permission from reference 75, P. H. Reddy, S. Das, D. Dutta, A. Dhar, A. V. Kir’yanov, M. Pal, S. K. Bhadra and M. C. Paul, Physica Status Solidi A 1700615 (2018), copyright @ 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

(NR-9200, EXFO, Canada). The average Numerical Aperture (NA) of the Er-doped nano-engineered fiber and Sc free standard Er-doped fiber (core diameter: 9.25 m) was estimated to be 0.18 and 0.19, respectively.

The Optical Performance of Erbium-Doped Nano-engineered SPYAS Glass-Based Optical Fiber Absorption spectra as shown in Fig. 14 were obtained using a white-light source with fiber output and an optical spectrum analyzer (OSA) with a 0.5 nm resolution. 3.0 cm length of fiber was used in actual experiment, and Fig. 14a shows the measured absorption (loss) spectra of Er-doped SPYAS glass-based optical fiber and Sc-free EDF where the bands peaking at 980 and 1530 nm adhere to the wellknown Er3C absorption transitions. On the other hand the absorption loss spectra of fiber drawn from the annealed SPYAS preform indicate multiple sharp peaks within the 980 and 1530 nm bands (Fig. 14b), which may signify that Er ions are present in environment of crystalline nanoparticles. Fluorescence spectra of the fiber samples were obtained by pumping at 980 nm wavelength under 250-mW pump power, in a lateral geometry. A fiber coupled mode pump laser diode at 976 nm is used as pump source. Such fiber pigtailed pump source was connected to the 980 nm arm of 980/1550 nm WDM coupler. A short piece of fiber around 4.5 cm having a peak small absorption less than 0.2 dB was fusion spliced to the 980/1550 nm arm of the WDM, its exit end was broken and immersed in an index matching fluid to avoid reflections. The fluorescence generated from the surface side of fiber collected at the WDM’s 1550 nm arm using the optical spectrum analyzer AQ-6315 (Ando) and was recorded using a computer attached to the OSA. An InGaAs photodetector was connected to record

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0.075 0.025 0.000 800

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a

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Fig. 14 Absorption spectra of the Sc-free standard Er-doped fiber/Er-doped pristine SPYAS glassbased fiber (a) and nano-engineered Er-doped SPYAS-based fiber (b). (Reprinted with permission from reference 75, P. H. Reddy, S. Das, D. Dutta, A. Dhar, A. V. Kir’yanov, M. Pal, S. K. Bhadra and M. C. Paul, Physica Status Solidi A 1700615 (2018), copyright @ 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

fluorescence decay time. The 976 nm laser diode was modulated externally by an acousto-optic modulator in the course of the lifetime measurement. We used less than 5 cm long fibers to avoid amplified spontaneous emission and reabsorption. The nano-engineered Er-doped SPYAS glass-based fiber shows a very broad fluorescence spectrum as well as strong emission signal at 1550 nm with increasing pump power at 980 nm compared to the Er-doped pristine SPYAS glass-based optical fiber and Sc-free standard Er-doped optical fiber shown in Figs. 15 and 16, respectively. It was observed that, with increasing pump power up to 350 mW at 980 nm, the intensity of 1.5-m emission increased significantly in the case of the nanoengineered SPYAS fiber, as evident from Fig. 16. The enhancement in fluorescence intensity in the annealed fiber is due to the presence of crystalline nanoparticles, where scandium (Sc) enhances the crystal field strength of Er ions as the ionic radius of Sc is smaller than that of Er ions (Fornasiero et al. 1998) and also, because of diminished up-conversion phenomena as yttrium (Y) cations substitute Er ions, preventing up-conversion involved neighboring Er ions due to similar ionic radius of Er and Y ions (Lo Savio et al. 2013). The fluorescence life time measurement curves of erbium-doped fibers drawn before and after thermal annealing were shown in Fig. 17. The fluorescence lifetimes of the 4 I13/2 ! 4 I15/2 emission of the pristine and annealed fibers are found to be 10.159 ms and 11.3 ms, respectively. The longer lifetimes of the 4 I13/2 levels together with the enhanced intensity of the 4 I13/2 ! 4 I15/2 emission in the annealed fiber sample compared to the base pristine EDF suggest a scandium environment for Er3C ions according to their incorporation in the nanocrystalline phase. Moreover, the maximum phonon energy of Sc2 O3 is reported around 625 cm1 (Palambo and Pratesi 2004). The low phonon energy along with the surrounding crystalline environment of erbium ions favors to enhance the fluorescence lifetime.

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Fig. 15 Fluorescence spectra of the Er-doped pristine, nano-engineered SPYAS glass-based optical fibers and Sc-free standard Er-doped optical fiber. (Reprinted with permission from reference 75, P. H. Reddy, S. Das, D. Dutta, A. Dhar, A. V. Kir’yanov, M. Pal, S. K. Bhadra and M. C. Paul, Physica Status Solidi A 1700615 (2018), copyright @ 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

1550 nm emission signal

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980 nm pump power (mW) Fig. 16 The effect of 980 nm pump power on 1550 nm emission signal intensity of the pristine SPYAS fiber, nano-engineered SPYAS fiber and Sc-free standard Er-doped optical fiber (c). (Reprinted with permission from reference 75, P. H. Reddy, S. Das, D. Dutta, A. Dhar, A. V. Kir’yanov, M. Pal, S. K. Bhadra and M. C. Paul, Physica Status Solidi A 1700615 (2018), copyright @ 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The experimental setup is shown in Fig. 18 to investigate the gain and noise figure characteristics of the fabricated SPYAS-based EDF. In the experiment, tunable laser source (TLS) as an input signal was varied from 1520 to 1620 nm wavelength. The variable optical attenuator (VOA) is used to obtain the specific and accurate input

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-1.0 y0 A1 t1

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-0.00218 0.00100 11.30311

±8.1224E-7 ±9.2325E-7 ±0.03112

ln(I/I0)

-1.4

B

-1.6 -1.8 -2.0 -2.2 0

5

10

15

20

25

30

35

Time, ms Fig. 17 Fluorescence decay curve of Er-doped nano-engineered SPYAS glass-based optical fibers drawn from the (a) pristine and (b) annealed preforms. (Reprinted with permission from reference 75, P. H. Reddy, S. Das, D. Dutta, A. Dhar, A. V. Kir’yanov, M. Pal, S. K. Bhadra and M. C. Paul, Physica Status Solidi A 1700615 (2018), copyright @ 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

power to the cavity. An isolator is placed after input power to prevent a backward ASE noise from entering the first stage and parasitic lasing which cause degradation of the optical gain of amplifiers. Ten meter lengths of the newly developed SYASbased EDFs drawn from both the pristine and annealed performs are used as a gain medium. The EDF is forward pumped by a 980 nm laser diode via a 980/1550 nm wavelength division multiplexing (WDM) coupler. The results were analyzed and measured by optical spectrum analyzer (OSA), which was located at the end of the configuration setup of the amplifier.

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40 35 30 25 20 15 10 5 0

14

Pristine SPYAS fiber Nano-engineered SPYAS fiber Er doped Sc free fiber

Noise Figure (dB)

Gain (dB)

Fig. 18 Experimental setup for measuring the gain and noise figure of SPYAS glass-based fiber. (Reprinted with permission from reference 75, P. H. Reddy, S. Das, D. Dutta, A. Dhar, A. V. Kir’yanov, M. Pal, S. K. Bhadra and M. C. Paul, Physica Status Solidi A 1700615 (2018), copyright @ 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

12

Normal SPYAS fiber Er doped Sc free fiber Nano-engineered SPYAS fiber

10 8 6 4 2 0

1520 1540 1560 1580 1600 1620

1520 1540 1560 1580 1600 1620

Wavelength (nm)

Wavelength (nm)

Fig. 19 Comparison of gain and noise figure spectra between Er-doped nano-engineered SYAS glass-based optical fiber drawn from pristine and annealed performs. (Reprinted with permission from reference 75, P. H. Reddy, S. Das, D. Dutta, A. Dhar, A. V. Kir’yanov, M. Pal, S. K. Bhadra and M. C. Paul, Physica Status Solidi A 1700615 (2018), copyright @ 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In Fig. 19, we show the gain and noise-figure (NF) spectra of 10-m pieces of the Er-doped nano-engineered SPYAS glass-based fibers, drawn from the pristine (red curve) and the annealed (black curve) preforms, measured under 175-mW laserdiode pumping at 980 nm using a  15-dBm signal as launch. As seen from the Fig. 19, the maximum 39.45 dB gain has been obtained at 1560 nm with the fiber drawn from the annealed preform. Furthermore, the nano-engineered optical fiber shows an average gain of 38.675 dB within the spectral gain flatness region. Such kind of nano-engineered optical fiber shows a better gain variation of ˙0.70 dB within 1530-1590 nm wavelength compared to the standard Sc-free Er-doped fiber, which is important for a broad band optical amplifier. On the other hand, the variation of the NF of the pristine Er-doped fiber, nano-engineered SPYAS glassbased fiber and standard Sc-free Er-doped fiber within the spectral gain flatness region is found to be (4.61–7.15), (4.35–6.85), and (5.5–7.67) dB, respectively, shown in Table 3. Figure 19 shows that the values of NF decrease from the nanoengineered SPYAG glass-based fiber to the standard Sc-free EDF.

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Table 3 Amplification parameters of Er-doped nano-engineered and printine SPYAS glass-based fibers along with Er-doped Sc-free standard fiber Amplification parameters Maximum gain at 1560 nm Flat-gain region Flat-gain(average value) Gain variation (dB) Noise fig. (NF) (dB)

Er-doped nano-engineered SPYAS fiber 39.45 dB

Er-doped normal SPYAS fiber

Er-doped Sc-free fiber

37.35 dB

35.89 dB

1530–1590 nm 38.675 dB ˙0.70 4.35–6.85

1530–1575 nm 36.765 dB ˙0.67 4.61–7.15

1530–1565 nm 36.5 dB ˙0.82 5.5–7.67

The improvement of gain performance in the first fiber is most probably explained due to the presence of annealing. Furthermore, the ligand field of the Sc subsystem induces the Stark effect because of site-to-site variations of the surrounding environment of Er ions due to inhomogeneous energy level degeneracy which may cause the widened optical transitions for getting broad band flat gain spectra. On the other hand, the passive loss of such kind of fiber arises from OH content giving rise to strong absorption at 1380 nm. Due to the presence of hydroxyl groups, energy transfer from 4 I13/2 level of Er3C to OH ions degrades the gain performance of the fibers. Here the energy levels of 4 I15/2 and 4 I13/2 are generally split more widely in SPYAS fiber drawn from annealed preform suggesting that the crystal field around Er ions sites is stronger in Sc2 O3 than in Y2 O3 due to the smaller ionic size of Sc3C than that of Y3C which bring the neighboring oxygens closer to Er3C surrounded by Sc2 O3 . The results suggest that such kind of Er-doped nanoengineered fiber-based amplifiers are expected to be useful in broadband optical communication systems.

Fabrication of Multielement (P-Yb-Zr-Ce-Al-Ca) Fiber for Moderate-Power Laser Application Fabrication of high power laser is an important research topic due to its wide range of applications considering materials processing, defense application, biomedical uses, etc. Although power level in the range of 1 kW had already achieved, researchers from different laboratories around the world are looking for higher power level for some specific application in the strategic field. Nevertheless, the bottleneck to achieve higher power is associated with nonlinear effects like SBS, SRS, four wave mixing, and photodarkening which must be controlled to get long term stability at high power level. Accordingly, it is inevitable to search for an improved glass host which could provide better thermal stability and thus accordingly we have presented a multielement (ME) fiber comprising Yb to test its performance under moderate power level.

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As mentioned earlier, the preform fabrication was carried out employing conventional MCVD-solution doping technique starting with Suprasil F-300 grade silica tube of dimension 20 mm (OD) with tube thickness of 1.5 mm. Different process parameters such as the porous core layer deposition temperature, solution composition, sintering environment, and temperature were precisely optimized in order to achieve the desired core glass composition. The process was initiated by deposition of numbers of matched cladding layers composed of pure silica at around 1850 ˙ 10 ı C. Subsequently, phosphosilicate porous layer was deposited at 1300 ˙ 10 ı C using the back-pass deposition technique to avoid complete sintering of the deposited soot layer. The presintering of the soot layer was one of the important process steps to achieve the desired dopant level in the final fiber, with temperature being precisely controlled and monitored using an IR pyrometer, moved synchronously with an oxy-hydrogen burner along the tube in forward direction. The porous soot layer was then soaked with an alcoholic solution comprising different dopants, namely AlCl3 .6H2 O (Alfa Aesar), YbCl3 .6H2 O (Alfa Aesar), CeCl3 .6H2 O (Alfa Aesar), ZrOCl2 .6H2 O (Alfa Aesar), and CaCl2 (Alfa Aesar), for fixed time span in different preform runs. The soaked tube was then dried under inert gas (nitrogen) flow and rejoined for subsequent processing. The soaked layer was then oxidized and dehydrated, followed by sintering employing the optimized conditions. The final preform was obtained through collapsing of the processed tube, followed by annealing of the preform at around 800–900 ı C for fixed time span prior to fiber drawing. Drawing of low RI coated fiber of bare diameter 125 ˙ 0.5 m was made from the fabricated preform after grinding to double D-shaped/hexagonal structure, employing a drawing tower (Heathway, UK). Hexagonal cladding has been achieved by grinding the initial fabricated preform equally from six phases followed by polishing prior to drawing of fiber. The inner cladding from flat to flat surface and vertex to vertex of hexagonal-shaped fiber is found to be 120 m and 130 m, respectively, while that of the outer polymer cladding is about 250 m. The core/clad boundary of the fabricated fibers was inspected using a high-resolution optical microscope. The use of different dopants in ME (P-Yb-Zr-Ce-Al-Ca) core-glass composition is to achieve superior properties due to change of local environment which can, in turn, tune spectroscopic properties. Al-doping is known to enhance RE solubility by providing a solvation shell (Aria et al. 1986), whereas P2 O5 doping helps in nucleation to growth of nano-crystals during annealing step (Varshneya (1994); Paul et al. 2012; d’Acapito et al. 2008; Rawson (1967)). CaO co-doping in multielement glass is known to stabilize Yb3C state (Singh et al. 2013; Sugiyama et al. 2013) and thereby reduces the formation of detrimental Yb2C ions which is one of the main reasons for the PD effect. Ca2C ion along with Al3C stabilizes Yb3C valence state with charge compensation in the silica glass. Ce2 O3 is known to prevent formation of Yb2C through charge transfer mechanism (Jetschke et al. 2016; Engholm et al. 2009) with the temporary oxidation of Ce3C to Ce4C which helps to trap the holes or electrons. The addition of ZrO2 plays significant role by providing thermal stability, thermo-chemical resistance, as well as optical transparency (Patra et al. 2002).

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Table 4 Properties of fibers under investigation ID

Core composition

NA (˙0.01)

Core diameter (˙0.1) m

A B

SiO2 -Yb2 O3 -P2 O5 -Al2 O3 SiO2 -Yb2 O3 -P2 O5 -Al2 O3 Ce2 O3 -CaO-ZrO2

0.15 0.15

10.4 10.3

Cladding absorption @ 976 nm, dB/m 7.9 8.3

Table 5 Comparison of elemental analysis of fibers under investigation ID A B

Yb (mol%) 05 06

Al (mol%) 19 18

P (mol%) 56 55

Ca (mol%) – 11

Ce (mol%) – 07

Zr (mol%) – 03

Amounts of the dopants incorporated in various preforms and their distributions across their cores were measured using a polished slice of thickness 1.5 mm using an electron probe microanalysis (EPMA). Formation of phase separated regions within the fiber core was detected using a high resolution transmission electron microscope (Technai G2 , Japan) along with measuring electron diffraction profiles in a selected area. The fiber parameters of standard phospho-alumino-silicate (PAS) glass-based Yb-doped fiber and multielements (ME) doped PAS glass-based fibers are given in Table 4. Table 5 presents the elemental composition of fabricated fibers. The main target was to investigate the role of multielements in standard phospho-aluminosilicate glass on lasing and photodarkening properties of the Yb-doped fiber laser where both the fibers maintained almost fixed Yb-content, as evident from their pretty close cladding absorption around 976 nm. It is also worth to note that both fibers also have approximately same P2 O5 and Al2 O3 content and thus any variation in the final performance of fiber B is due to addition of Ca, Ce, and Zr in glass structure. The measurements of absorption and fluorescence spectra, using an optical spectrum analyzer (OSA) of the Ando type, as well as the measurements of fluorescence lifetimes and analysis of the resultant fibers in the sense of highpower applications, were made. The absorption spectra were measured applying a standard cut-back procedure with the use of a white-light source; Yb fluorescence spectra and kinetics were obtained in-cladding pumping of the fibers by a pig-tailed diode laser at 976-nm wavelength decay in the lateral detecting arrangement. The Yb-doped fibers were tested under the laser configuration, schematically presented in Fig. 20. A double D-shaped fiber in double clad structure was pulled in 125 m inner cladding diameter. The large inner cladding diameter was chosen to enable an efficient pump launch from the high-power pump laser diodes. The fiber was end-pumped by a 976 nm laser diode. Pump launch end of the fiber was cleaved perpendicular to the fiber axis to provide 4% Fresnel reflection to form the laser cavity. At the other end, a high reflective mirror (100%) at signal band was used to close the laser cavity.

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Fig. 20 Schematic of laser setup. (Reprinted with permission from reference 2, A. Dhar, M. C. Paul, S. Das, H. P. Reddy, S. Siddiki, D. Dutta, M. Pal and A. V. Kir’yanov; Physica Status Solidi A, 214, 4–7 (2017), copyright @ 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Lock in amplifier

InGaAs detector Monochromator

Lens

Computer

Active Yb doped fiber

Chopper

Lens He-Ne

Splicing point

LD

Fig. 21 Schematic of experimental PD setup. (Reprinted with permission from reference 2, A. Dhar, M. C. Paul, S. Das, H. P. Reddy, S. Siddiki, D. Dutta, M. Pal and A. V. Kir’yanov; Physica Status Solidi A, 214, 4–7 (2017), copyright @ 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The PD of the fibers was evaluated by monitoring the transmitted probe power at 633 nm through the test fiber under 976 nm irradiation. The PD measurement setup is presented in Fig. 21. A fiber-coupled single-mode 976 nm laser diode was used as a pump source. The output end of the pump fiber was spliced to one port of a wavelength-division multiplexing (WDM) coupler, and the pump beam was delivered to the test fiber by splicing the output end of the WDM coupler and the test fiber. A He-Ne laser at 633 nm was used as a probe beam once coupled to the test fiber through the WDM. The probe beam propagated in the same direction as

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the pump beams. The output end of the test fiber was spliced to another WDM coupler to separate the pump and probe beam. The probe beam was chopped by a mechanical chopper, and the output power was detected by a photo-detector and lock-in amplifier after passing through a monochromator. We used very short piece (1–2 cm) of the test fiber to avoid any effect of amplified spontaneous emission on the data and to diminish possible re-absorption increasing at PD. The pump power was maintained to provide 40% of population inversion of Yb3C ions along the test fibers.

Material Characterization of Multielement (P-Yb-Zr-Ce-Al-Ca) Optical Preform and Fiber Refractive index profile of different preform/fiber samples was found to have a central dip, especially featured in GeO2 and P2 O5 doped performs fabricated using the MCVD process. An example of RI profile (RIP) of fiber sample B is presented in Fig. 22 which reveals no central dip which is achieved due to use of multielement glass host composition. The analysis of core sample under high resolution transmission electron microscope (HRTEM) revealed that phase-separation occurs in preform sample B but not in A. Presence of liquid miscibility gap in the phase diagram of our selected glass composition is the main reason behind achieving the phase-separation within the core area. Interestingly, nano-particles within the phase-separated core region

0.02

0.015

Del n

0.01

0.005

0 –120

–80

–40

0

40

80

120

–0.005

Distance (Micron) Fig. 22 Refractive index profile of fiber B. (Reprinted with permission from reference 2, A. Dhar, M. C. Paul, S. Das, H. P. Reddy, S. Siddiki, D. Dutta, M. Pal and A. V. Kir’yanov; Physica Status Solidi A, 214, 4–7 (2017), copyright @ 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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Fig. 23 HRTEM analysis data with ED pattern of ME-Yb fiber (sample B) showing particles of size range 7–10 nm. (Reprinted with permission from reference 2, A. Dhar, M. C. Paul, S. Das, H. P. Reddy, S. Siddiki, D. Dutta, M. Pal and A. V. Kir’yanov; Physica Status Solidi A, 214, 4–7 (2017), copyright @ 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

retained only in sample B in its fiber stage. This clearly indicates better thermal stability of core glass composition in preform sample B and the effect of addition of ZrO2 . The HRTEM data of annealed fiber sample B are presented in Fig. 23 (main frame) together with the selected area electron diffraction (SAED) pattern (inset) which indicates the phase-separated nano-particles ranging from 7 to 10 nm with partially crystalline nature. The electron diffraction X-ray (EDX) analysis associated with HRTEM confirms the presence of Zr ion in the fiber core. The elemental distribution of different dopants in the fiber core of fiber sample B measured using EPMA is presented in Fig. 24 while for fiber sample A is presented in Fig. 25. The result obtained reveals that there is no central dip formation in case of P2 O5 which is generally observed in P2 O5 -doped fiber sample prepared through the MCVD process. This type of dopant distribution curve was observed in our fabricated ME optical fibers without any depression of P2 O5 level and probably occurred due to suppression of evaporation of P2 O5 in the presence of Zr, Ca and Ce. The spectra of absorption loss were obtained using a white-light source through Bentham spectral attenuation setup with a fiber output and optical spectrum analyzer (OSA) turned to a 0.5 nm resolution. The spectrum for fiber sample B was compared with standard PAS fiber (sample A) to understand the effect of different dopants (Ca, Zr, and Ce) comprising of similar Yb2 O3 concentration, 0.15 mol%. The combined data presented in Fig. 26 clearly reveal that although the 975-nm peak height is virtually the same, there is a variation in the shoulder around 915-nm and this variation is the characteristic of the core glass composition.

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Fig. 24 Dopants distributions in ME-Yb fiber (sample B) measured using EPMA. (Reprinted with permission from reference 2, A. Dhar, M. C. Paul, S. Das, H. P. Reddy, S. Siddiki, D. Dutta, M. Pal and A. V. Kir’yanov; Physica Status Solidi A, 214, 4–7 (2017), copyright @ 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

7 Al2O3

6

Oxide (mol %)

Fig. 25 Dopants distributions in standard Yb fiber (sample A) measured using EPMA. (Reprinted with permission from reference 2, A. Dhar, M. C. Paul, S. Das, H. P. Reddy, S. Siddiki, D. Dutta, M. Pal and A. V. Kir’yanov; Physica Status Solidi A, 214, 4–7 (2017), copyright @ 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

P2O5

5

Yb2O3

4 3 2 1 0 0

10

20

30

40

Fiber diameter (Micron)

The Optical Performance of Multielement (P-Yb-Zr-Ce-Al-Ca) Optical Fiber To study the lasing performance, we have measured the output power using the experimental setup described earlier in Fig. 20. It is clearly observed that the output power linearly increased with the launched pump power. The output power reached 37.5 W at launching 50 W of pump power, representing good slope efficiency of 76% with respect to the launched pump power. The comparison of output powers obtained from standard fiber A and ME fiber B is presented in Fig. 27. The output

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Cladding absorption (dB/m)

8

ME-Yb Stand Yb

6 4 2 0 800

900

1000

1100

Wavelength (nm) Fig. 26 Comparison of cladding absorption of standard phospho-alumino-silicate Yb fiber (sample A) with ME Yb fiber (sample B). (Reprinted with permission from reference 2, A. Dhar, M. C. Paul, S. Das, H. P. Reddy, S. Siddiki, D. Dutta, M. Pal and A. V. Kir’yanov; Physica Status Solidi A, 214, 4–7 (2017), copyright @ 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) Fig. 27 Laser output power of standard phospho-aluminosilicate Yb fiber (sample A) and ME Yb fiber (sample B). (Reprinted with permission from reference 2, A. Dhar, M. C. Paul, S. Das, H. P. Reddy, S. Siddiki, D. Dutta, M. Pal and A. V. Kir’yanov; Physica Status Solidi A, 214, 4–7 (2017), copyright @ 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

power in both A and B reported here is not the maximum but can be further enhanced by increasing launched pump power. In general, the induced photodarkening (PD) loss is proportional to the inversion level of the Yb3C ions. However, it is well known fact that the selected host material can influence PD significantly; e.g.; phosphosilicate glass is known to lead to suppression of PD loss in a significant amount, as compared to the aluminosilicate counterpart. The PD behavior of both fibers was thus carried out to evaluate the effect of host-glass composition with special emphasis on effect of dopant ions like Ca, Ce and Zr ions present in fiber B. The small signal absorption in different fibers was around 7–8 dB/m. The temporal characteristics of the transmitted probe power @633 nm are represented in Fig. 28 for standard Yb-doped fiber with our fabricated

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Transmitance at 633nm (%)

Fig. 28 PD induced loss of standard phospho-aluminosilicate Yb fiber (sample A) and ME Yb fiber (sample B). (Reprinted with permission from reference 2, A. Dhar, M. C. Paul, S. Das, H. P. Reddy, S. Siddiki, D. Dutta, M. Pal and A. V. Kir’yanov; Physica Status Solidi A, 214, 4–7 (2017), copyright @ 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

511

Stand Yb ME-Yb

100 80 60 40 20 0 0

20

40

60

80 100 120 140 160 180 200

Time (minutes)

ME-Yb-doped fiber. The observed result clearly indicates that the PD induced loss is significantly reduced in the Yb-doped ME (P2 O5 -ZrO2 -Al2 O3 -CeO2 -CaO) fiber. A relation has been established between the absorption coefficient change of probe power @633 nm and the time under pumping at 976 nm. The experimental curves are fitted by exponential decay curve (red color) shown in Figs. 29 and 30 where the ’(t) is the absorption coefficient change in units of dB/m, the constant ’sat is the saturation parameter and £ is the time constant. The following relation is given below with time. ’ .t/ D ’sat C ’sat  e.t=£/ : The photodarkening-induced absorption coefficient change at 633 nm for standard Yb-doped fiber (A) is found to be very fast compared to multielements glass-based Yb-doped fiber (B) as presented in Figs. 29 and 30. The photodarkening phenomenon of Yb-doped fibers show spectrally broad transmission loss centering at the visible wavelengths and extending up to the pump and signal wavelength region due to formation or creation of color centers which causes the main absorption band at visible region tailoring to the NIR region. Therefore, monitoring of transmitted power of 633 nm visible light with time under pumping at 976 nm will give an indication about the photodarkening behavior of Yb-doped fibers. So, we have monitored the transmitted probe power of 633 nm visible light with time (Koponen et al. 2006). In case of high power operation of fiber laser, there is a probability of reduction of Yb3C to Yb2C ions with formation of free holes. These holes then get trapped at various defects and imperfections in the glass structure with the result of an induced absorption. At the same time, Yb2C ions have possibility to oxidize and transform back to Yb3C ion with the release of an electron. To overcome this problem, Ce, Ca, and Zr are added into the doping host of Yb whose already explained in the fabrication part.

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0

–Δα633nm (dB/m)

–50 –100

Parameters

–Δα(t) = Δαsat – Δsat.e–(t/τ)

Δαsat τ

351.5

Adj. R-square Reduced Chi-Sqr.

0.99535 44.49592

19.41473

–150 –200 –250 –300 –350 0

20

40

60

100

80

120

140

Time (minutes) Fig. 29 Photodarkening induced absorption coefficient change ’ of standard fiber A at 633 nm with time at 10.0 W pump power. (Reprinted with permission from reference 2, A. Dhar, M. C. Paul, S. Das, H. P. Reddy, S. Siddiki, D. Dutta, M. Pal and A. V. Kir’yanov; Physica Status Solidi A, 214, 4–7 (2017), copyright @ 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

–Δα(t) = Δαsat – Δsat.e–(t/τ)

Parameters 0

Δαsat τ

–Δα633nm (dB/m)

–50

250.9629

Adj. R-square Reduced Chi-Sqr.

34.65149 0.99654 18.99509

–100 –150 –200 –250 0

20

40

60

80

100

120

140

Time (minutes) Fig. 30 Photodarkening induced absorption coefficient change ’ of multielement glass-based fiber B at 633 nm with time at 10.0 W pump power. (Reprinted with permission from reference 2, A. Dhar, M. C. Paul, S. Das, H. P. Reddy, S. Siddiki, D. Dutta, M. Pal and A. V. Kir’yanov; Physica Status Solidi A, 214, 4–7 (2017), copyright @ 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In support of our observation regarding PD experiment, we compared the peak originated around 220–230 nm associated with the formation of Ytterbium Oxygen Defect Center (YbODC) (Yoo et al. 2007; Engholm and Norin 2008). The peak intensity related to YbODC was found to reduce appreciably in our ME-Yb-doped

Fig. 31 Comparison of transmission loss in UV region between standard phospho-aluminosilicate Yb-fiber (sample A) with ME-Yb fiber (sample B). (Reprinted with permission from reference 2, A. Dhar, M. C. Paul, S. Das, H. P. Reddy, S. Siddiki, D. Dutta, M. Pal and A. V. Kir’yanov; Physica Status Solidi A, 214, 4–7 (2017), copyright @ 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Absorption Coefficient (cm–1)

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2.5 2.0 Stand Yb-doped fiber

1.5

ME-Yb doped fiber

1.0 0.5 0 200

220

240

260

280

300

Wavelength (nm)

optical fiber sample with respect to standard alumino-phospho silicate glass-based optical fiber (A), as evident from Fig. 31. Ytterbium-doped alumina-silica glass has a strong absorption band near 230 nm, known as a charge transfer band. This absorption band corresponds to the transfer of an electron from the nearby ligands (oxygen) to the ytterbium ion with the formation of a divalent Yb ion and a localized hole left behind on the surrounding ligands. The probability for the creation of mobile charges decreases with addition of ZrO2 in the glass structure and hence reduces the formation of induced color centers. This fact indicates that the presence of ME core glass reduces the rate of formation of YbODC centers. PD in the Yb-doped aluminosilicate takes place through the breaking of ODCs under two-photon absorption which gives rise to release of free electrons. The released electrons may then be trapped at Al or Yb sites and form a color center resulting in PD. To avoid that situation, we have selected ME core glass to reduce the formation of YbODC significantly. Additionally, formation of nano-phase separated region is expected to facilitate shifting of a lone electron to the nearest neighbor nonbridging oxygen (NBO) atom bonded to Yb ions into the positive charge vacancy zones under high-pump power and therefore enhances PD resistivity significantly. As the field strength of the modifier cation enhances either through decreasing ionic radius or increasing valence charge, which may give the perturbation of aluminosilicate network more strongly. Such perturbation occurs because of the energetically stabilization of Yb ions provided by closer association of negatively charged species, in particular NBO.

Fabrication of Chromium-Doped Nano-phase Separated Yttria-Alumina-Silica (YAS) Glass-Based Optical Fiber Chromium (Cr)-doped optical fiber is very important in terms of its application as saturable absorber and in optical coherence tomography (OCT) covering a wavelength region of 1.1–1.6 m range. Nevertheless, it is important to stabilize

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Fig. 32 Appearance of yellow coloration during oxidation stage after solution soaking. (Reprinted with permission from reference 17, D. Dutta, A. Dhar, A. V. Kir’yanov, S. Das, S. Bysakh and M. C. Paul; Phys. Status Solidi A 212 1836 (2015), copyright @ 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Cr4C inside the suitable glass host structure to achieve the abovementioned broad band fluorescence. Although there are some literature available which deals with silica glass-based Cr4C -doped fibers based on alumino-silica, germano-silica, and gallium-silica glasses, fabricated using the well-known MCVD-solution doping (SD) technique but nobody able to report broadband emission of Cr4C ions unlike bulk glass systems as it is extremely difficult to stabilize Cr4C in silica glass-based optical fiber. Accordingly, here we have presented fabrication of nanophase separated yttria-alumina-silica glass-based optical fibers through the MCVD process in conjunction with a solution doping technique under suitable thermal annealing conditions. Chromium-doped preform fabrication process starts with the deposition of porous silica layers formed due to oxidation of SiCl4 vapor in presence of oxygen inside a waveguide silica tube of an outer dimension sized by 20 mm with thickness of 1.5 mm at a suitable temperature (Fig. 32). The temperature during deposition was monitored using an IR pyrometer which moves synchronously with an oxyhydrogen burner along the tube in forward direction. Different dopant ions, namely, Al, Cr, Mg, and Y, were introduced into the porous frit employing the SD technique (Townsend et al. 1987).

Material Characterization of Chromium-Doped Nano-phase Separated YAS Glass-Based Optical Preform and Fiber The concentrations of the dopants inside the cores were varied by means of altering the composition of soaking solution comprising AlCl3 , 6H2 O (Alfa Aesar), CrCl3 , 6H2 O (Alfa Aesar), MgCl2 , 6H2 O (Alfa Aesar) and Y(NO3 )3 , xH2 O (Alfa Aesar), dissolved in a mixture of water and ethanol. After soaking for a stipulated time span, the soaked tube was air dried, remounted on the glass working lathe followed by subsequent processing to obtain the final fiber. Refractive index profiles (RIP) of the fabricated preforms and fibers were measured using preform analyzer (PK-2600, Photon Kinetics, USA) and a fiber

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Table 6 Details of representative fibers Fiber no. Cr-2 HCr-2

Core composition SiO2 -Al2 O3 -MgO-Y2 O3 -Cr2 O3 SiO2 -Al2 O3 -MgO-Y2 O3 -Cr2 O3

NA ˙0.01 0.21 0.21

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analyzer (NR-9200, EXFO, Canada). The amounts of incorporated dopants and their distributions across the cores were measured using an electron probe microanalysis (EPMA) using a polished sample of thickness 1.5 mm. The microstructure of the porous deposit was evaluated using a scanning electron microscope (SEM) while in the fiber was measured using a high resolution transmission electron microscope (HRTEM) (Technai G2 , Japan) along with selected area electron diffraction (SAED) profile. The absorption spectra of the fabricated fibers in the ranges of 300– 1100 nm and 400–1600 nm were measured using a Bentham setup (UK) and an Optical Spectrum Analyzer (OSA) (AQ6370B, Yokogawa, Japan), respectively. The emission spectra of samples with thicknesses of 1 cm were measured by exciting the samples at 370 and 474 nm using a Xenon lamp (Edinburgh Instrument, UK). Accordingly, the fluorescence spectra of fiber samples were measured using the OSA at excitation by an Ytterbium fiber laser (wavelength 1064 nm, IPG Polus, USA), in frontal geometry. This characterization method was repeated for different fiber samples before and after annealing to investigate its effect with respect to changes in concentration of chromium. The heat-treated fibers are assigned by the symbol “HCr.” The annealing conditions (temperature, time span, and heat cycles) were optimized based on our initial characterization results. The detailed results obtained using one representative fiber sample is presented here (Table 6). The average numerical aperture (NA) of the fabricated Cr-2 was found to be around 0.21 with good uniformity along its overall length (20 cm) and the representative RIP profile of the core region is demonstrated in Fig. 33. The dopants distributions measured using EPMA inside the fiber core are presented in Fig. 34a and b which reveal quite uniform profiles with concentrations of the dopants being 8.5 wt% (Al2 O3 ), 0.75 wt% (Y2 O3 ), 0.013 wt% (Cr2 O3 ), and 2.0 wt% (MgO). It is worth noting here that a divalent co-dopant (Mg2C in our case) has to be incorporated for generating tetrahedral coordination of chromium as it facilitates stabilization of the Cr4C species. The nature and the size of the phase-separated particles within the core area (annealed and un-annealed) as well as in the final fiber samples were studied using the HRTEM and SAED analyses. The TEM image of the active core glass with ED pattern for Cr-2 nonannealed preform sample is shown in Fig. 35a. The HRTEM image of Cr-2 reveals absence of any phase-separated regions, whereas the HRTEM image of the fiber drawn from same preform after annealing (HCr-2) at 1200 ı C exhibits presence of nano-phase separated inclusions, as evident from Fig. 35b. This nonuniformity reduces to below 5 nm in the case of the fiber drawn from the annealed preform with significant reduction in nonhomogeneity in the core glass shown in Fig. 36b. Since there is no phase separation in the nonannealed preform,

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Fig. 33 RIP profile of chromium-doped fiber Cr-2. (Reprinted with permission from reference 17, D. Dutta, A. Dhar, A. V. Kir’yanov, S. Das, S. Bysakh and M. C. Paul; Phys. Status Solidi A 212 1836 (2015), copyright @ 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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Fig. 34 (a) Distributions of different dopants (Al2 O3 , MgO, and Y2 O3 ) across the core area of fiber Cr-2 and (b) distribution of Cr2 O3 across the core area of fiber Cr-2. (Reprinted with permission from reference 17, D. Dutta, A. Dhar, A. V. Kir’yanov, S. Das, S. Bysakh and M. C. Paul; Phys. Status Solidi A 212 1836 (2015), copyright @ 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

the dark boundary surrounding the bright circle in the SAED pattern is not so prominent. The phase-separated inclusions appear as black spots with average size ranging from 10 to 60 nm. The corresponding size distribution for nano-phase in the 30–60 nm range is presented in Fig. 36b (average size 20–30 nm). The spot EDX on the globule and the matrix clearly indicates that the phase-separated nano-phase is rich in Al and Y, thus indicating the composition is possibly YAS glass. In case of the fiber drawn from the annealed samples as in Fig. 35b, those inclusions transform into very fine spots that tend to dissolve into the parent glassy phase with average

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Fig. 35 TEM pictures along with ED pattern of chromium-doped glass-based (a) nonannealed preform (Cr-2) and (b) fiber drawn from the annealed (HCr-2) preform. (Reprinted with permission from reference 17, D. Dutta, A. Dhar, A. V. Kir’yanov, S. Das, S. Bysakh and M. C. Paul; Phys. Status Solidi A 212 1836 (2015), copyright @ 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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Fig. 36 TEM pictures along with ED pattern of chromium-doped glass-based preform: (a) annealed preform (HCr-2) and (b) particle size distribution in annealed preform (HCr-2). (Reprinted with permission from reference 17, D. Dutta, A. Dhar, A. V. Kir’yanov, S. Das, S. Bysakh and M. C. Paul; Phys. Status Solidi A 212 1836 (2015), copyright @ 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

size around 5 nm shown in Fig. 36a. A possible reason of decrease in particle size is re-melting and stretching of the separated phase during the fiber’s drawing. The SEM images of porous structure before and after solution doping (SD) were examined in order to get an idea about nonuniformity in the dopants distributions in the final core. Additionally, a comparison of the EDX analysis of porous soot after making SD with that of the final one reveals that significant amount of chromium ions was lost due to evaporation during sintering/collapsing. The colorless soot

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particles during drying in the presence of oxy-hydrogen flame (tube’s outside temperature 800 ı C) first changes to yellow color as seen from Fig. 32 (due to the formation of Cr(VI) species) and is converted into clear glass after complete sintering. We observed faint greenish color inside the core for Cr2 O3 concentration above 0.15 wt%. Due to the small size of nano-phases observed at the fiber stage (Fig. 36b), a spot EDX analysis could not be performed and so the exact composition of these nanosize phases could not be determined. It can only be proposed that those granules with dark appearance may possess elements having higher atomic mass. According to our earlier work (Halder et al. 2013) on Tm-Yb co-doped nano-phase separated yttrium-alumino-silicate (YAS) core glass fibers, it was found that most of Tm-Yb ions concentrate within the black granular nano-YAS phases and that those granular nano-YAS phases are composed of Al2 O3 , Y2 O3 , and SiO2 along with REs (Tm and Yb). Here we observed similar black granular phase separation and due to the similarity of the two glasses, we can presume that the black spots are chromium-rich zones within YAS glass which also observed in the case of the RE-doped nano-YAS matrix (Halder et al. 2013).

The Optical Performance of Chromium-Doped Nano-phase Separated YAS Glass-Based Optical Fiber The absorption spectra of the fibers drawn from preforms Cr-2 (nonannealed) and HCr-2 (annealed) were measured using short (a few to tens cm) lengths of fiber; the resultant spectra are shown in Fig. 37. Both spectra can be equally decomposed into at least four bands in the VIS and near-IR, peaked at 430 (I), 630 (II), 860 (III), and 1030 (IV) nm and additionally an intensive band (V) in the UV within 300– 400 nm is present in the spectra (inset to the figure). Based on literature data (Wang and Shen 2012; Murata et al. 1997), the absorption bands of both Cr-2 and HCr-2 fibers having Mg and Cr-doped alumino-silica glass is given below. The bands I and II are assigned to the (d-d) transitions 4 A2 -4 T1 and 4 A2 -4 T2 , 4 A2 -2 T1 , and 4 A2 -2 E of Cr3C ions in octahedral coordination, while band V to the charge-transfer transition of Cr6C ions in octahedral coordination. The bands III and IV generate due to the presence of Cr4C in tetrahedral coordination with transitions 3 A2 -3 T1 and 3 A2 -3 T2 . In the meantime, the first of these transitions covers partially band II; so this spectral band (centered at 630 nm) seems to comprise the contributions from both Cr4C (in the most degree) and Cr3C (in the least degree) ions. It is found from Fig. 37 that although the basic nature of the absorption spectra did not change after annealing, the extinctions of all the absorption bands in VIS to near-IR (I to IV) decrease. The same phenomenon happened for the band V located in the UV (compare curves 1 and 2). This clearly points out that certain structural rearrangements have occurred inside the fiber as a result of thermal treatment. Since the drop of extinction in all the bands’ maxima is virtually equal and measured to be 3.0–3.5 times (notice that the band widths do not demonstrate changes), we can conclude that no serious transformations among Cr4C , Cr3C , and Cr6C ions

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Fig. 37 Absorption spectra of chromium-doped fiber samples drawn from the preforms (a) before (Cr-2, curve 1) and (b) after (HCr-2, curve 2) annealing. Dashed lines show the bands I to IV that fit the experimental spectra 1 and 2 (a deconvolution procedure using Gaussian shape of the bands was used). (Reprinted with permission from reference 17, D. Dutta, A. Dhar, A. V. Kir’yanov, S. Das, S. Bysakh and M. C. Paul; Phys. Status Solidi A 212 1836 (2015), copyright @ 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

happened (Dvoyrin et al. 2003), but rather a kind of interdiffusion process (Huang et al. 2013) of the dopants (including chromium ions) and Si (from the adjacent cladding region) took place at thermal annealing with the result being overall drop of extinction in all bands I to V. Another reason behind the phenomenon could be “generation” of different chromium species in lower valences (C1 and C2) at annealing, but the available literature data provide no evidence for such kind of process to go further. Furthermore, an explanation behind negligible conversion of Cr4C species to Cr3C at the thermal treatment that we applied seems to be a stabilizing effect, given by the presence of Mg in the core glass network, which is a species mainly involved Cr4C ions formation. At the same time, the partial re-structuration of the core glass occurs from nonannealed to annealed stage of optical preform Cr-2—HCr-2 through suitable thermal annealing process. One of the consequences of the latter process is demonstrated by an experiment on excitation of fluorescence in Cr-2 and HCr-2 fibers at 474-nm pumping. The result is presented in Fig. 38. The emission spectra at 474-nm excitation of the fibers exhibited a complex, but qualitatively similar to each other, spanning 500 to 800 nm spectral range. This fluorescence is associated with the presence of Cr3C ions, as already reported in literature (Murata et al. 1997). However, its intensity was found to significantly

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Fig. 38 Emission spectra of chromium-doped fiber samples before (Cr-2) and after (HCr-2) annealing process under pumping at 474 nm wavelength. In inset showing the enlarged view of emission spectra. (Reprinted with permission from reference 17, D. Dutta, A. Dhar, A. V. Kir’yanov, S. Das, S. Bysakh and M. C. Paul; Phys. Status Solidi A 212 1836 (2015), copyright @ 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

enhanced under optimized annealing around 1200 ı C (i.e., in sample HCr-2), as evident from Fig. 38. We guess that the main cause of the fluorescence enhancement effect is nano-phase re-structuration of the core glass (taking place as a result of thermal treatment), positively affecting the fluorescent ability of the emitter, in this case being Cr3C ions. Quite short pieces of fibers (2.5 and 5.5 cm) were used in this experiment, providing 2.5–3.0 dB absorption of pump light. The spectra of fluorescence spanning from 900 to 1400 nm (centered at 1100 nm) were detected in both cases in vastly the same experimental conditions and so are worth of direct comparison. We present here the data obtained at 50 mW pump power launched into the fibers (the spectrum of pump is shown by curve 3). Cr4C fluorescence spectra recorded at higher pump powers (up to 2.5 W) revealed almost no change in shape and negligible rise of intensity, given that saturating of fluorescence power happens at very low pump power, measured by 5–10 mW. As seen from Fig. 39, the fluorescence spectra are quite similar for Cr-2 and HCr-2 fibers, which shows that annealing does not lead to any pronounced worsening of the fluorescent ability of Cr4C after thermal treatment. Emphasized that the fluorescence spectra in Fig. 39 match well the ones reported earlier for other types of Cr, Mg co-doped aluminasilica glass and fiber, also ascribed to Cr4C ions (Huang et al. 2010, 2013; Wang and Shen 2012). Let’s consider the results of an experiment demonstrating the bleaching effect in Cr-2 and HCr-2 fibers under the action of 1.064-m pump light as shown in Figs. 40a, b, respectively. In the experiment, we employed the same setup as was used in the fluorescence spectra measurements (Fig. 39), but now we measured

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Fig. 39 Emission spectra of chromium-doped fiber samples drawn from the preform before (Cr-2, curve 1) and after (HCr-2, curve 2) annealing. The spectrum of pump light is given in (a) by curve 3 (pump power is 50 mW in all cases). œc marks the central wavelength of the emission spectra. (Reprinted with permission from reference 17, D. Dutta, A. Dhar, A. V. Kir’yanov, S. Das, S. Bysakh and M. C. Paul; Phys. Status Solidi A 212 1836 (2015), copyright @ 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

transmission coefficients (T) of pump light with power P after propagation through pieces (a few to tens cm in length L) of Cr-2 and HCr-2 fibers; these data were then proceeded for getting the nonlinear (dependent on incidence pump power) “resonant” absorption coefficients ˛ of the fibers, applying formula ˛(P) D ln[T(P)]/L (i.e., the quantities provided in Fig. 40) in function of P. It is seen from the figures that in both fibers a pronounceable bleaching effect, viz., a decrease of absorption against pump power is observed. Moreover, the effective saturating (bleaching) pump power (Psat ) is measured by a few (5–10) mW only, which is quite promising for utilizing of these or similar chromium-doped fibers as a Q-switch fiber for near-IR pulsed applications. Given that core diameter of both Cr-2 and HCr-2 fibers is 10 m (Table 6), the saturating intensity at 1.06 m is estimated to be around 10 kW/cm2 . Furthermore, if one assumes that the decay of the near-IR fluorescence of Cr4C ions (Fig. 39) is around 10 s, the ground-state absorption cross-section of Cr4C ions in the fibers at 1.0–1.1 m (i.e., via transition 3 A2 -3 T2 band IV; Figure 40) can be estimated to be 1018 cm2 . Such value is comparable with the ones reported elsewhere (V. Felice et al. 2000) for chromium-doped alumina-silica glasses. Finally, a brief discussion on the backgrounds we employed to make the fibers ought to be highlighted. As chromium is doped into YAS phase-separated particles through the solution doping technique, it substitutes aluminum and normally

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Fig. 40 Dependences (points) of resonant absorption coefficient a of Cr-2 (a) and HCr-2 (b) fibers upon 1064 nm pump power P. Squares (at P D 0) mark the small-signal absorption values (a0 ) at 1064 nm, also denoted in insets where near-IR part of the fibers absorption spectra are reproduced (Fig. 5). Psat are the saturation pump power at which a is decreased twice relatively to a0 values. (Reprinted with permission from reference 17, D. Dutta, A. Dhar, A. V. Kir’yanov, S. Das, S. Bysakh and M. C. Paul; Phys. Status Solidi A 212 1836 (2015), copyright @ 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

occupies the octahedral site, which is seen as the characteristic band in Fig. 37, ascribed to Cr3C species. On the other hand, when the divalent co-dopant Mg2C is introduced into glass, it substitutes Y3C in dodecahedral site and leaves a negatively charged vacancy in the lattice. During the sintering process of soaked layer under slightly reducing environment, the oxygen-excess defect centers (such as super oxide ion O2  and peroxy bonding –O–O–) serve as an oxidizing agent for conversion Cr3C ! Cr4C . The formation of these oxygen-excess defect centers can be generated in the presence of a large content of glass modifier cations such as Mg. Cr4C has a larger ionic size (0.041 nm) than Si4C (0.026 nm) in coordination 4 and may be associated with bridging and/or nonbridging oxygen of the silica network (Shannon 1976).

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However, in solid state oxides of aluminum, yttrium, and chromium cations are incorporated more preferably in the C3 oxidation state in an octahedral environment of the nearest oxygen atoms. Similarly, in the silica glass network that contains aluminum, chromium atoms can replace silicon atoms and be stabilized in the tetrahedral environment. During this transformation aluminum atoms form negatively charged complexes [AlO4/2 ]2 (Murata et al. 1997). In silica glass, chromium ions presented in the second coordination sphere of aluminum and yttrium ions can serve as charge compensators which contribute to stabilization of aluminum in the tetrahedral site. At the same time, such charge compensation also favors stabilization of Cr4C in the metastable tetrahedral site (Murata et al. 1997). Such a situation arises during the fiber drawing process with fast glass quenching. Thus, the effects of co-doping core glass with Al and Mg upon formation and stabilization of Cr4C ions (and also upon hardening against decay to Cr3C ions at thermal annealing) seem to be equally positive. Presently, we are targeting at optimizing the chromium-doped fabrication conditions and also the core glass composition in order to achieve stable Cr4C species in higher concentrations in the fiber stage and, thereafter, to get broader near-IR fluorescence and the bleaching effect (Fig. 40) of a higher contrast at pumping at 1.06 m or at near wavelength.

Conclusions In conclusions, we have focused the basic material and optical properties of advanced nano-engineered glass-based optical fibers doped with erbium/ytterbium, erbium/scandium, ytterbium, and chromium for photonics applications such as multichannel optical amplification in C-band, broadband optical amplification in the CC L band region, fiber laser with enhanced PD resistivity, and broad-band source at NIR. A new class of optical fiber based on Er-doped crystalline zirconia yttria alumina phospho silica nano-particles made through a solution doping technique followed by a MCVD process with optimization of the process parameters at different stages of fabrication. Glass modifiers and nucleating agents are added into the host glass followed by a proper annealing of the preform to generate uniform distribution of crystalline zirconia yttria alumina-phospho silica nanoparticles in the core region of optical fiber. Such new type of Zr-EDF was analyzed as multichannel amplification in the C-band region where this particular fiber exhibits maximum gain difference 22 dB at the input signal level of 30 dBm/channel amplification under pumping at 980 nm wavelength. Another novel material design-based optical fiber developed through incorporation of Er ions into nano-engineered scandium-yttria-alumina-silica (SYAS) glass through the MCVD process coupled with the solution doping technique and followed by thermal annealing of the fabricated perform in order to enhance the fluorescent ability along with optical gain in the C- and L-bands. Such kind of

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phenomenon have been established owing to the suitable thermal annealing of the preform, as compared to the one drawn from pristine (not-annealed) preform. The absorption spectra along with the SAED patterns from the TEM analyses suggest that the revealed enhancements in the fiber drawn from the annealed preform occurs through the modification of surrounding environment of Er3C ions within or in proximity to the crystalline nanoparticles which generated during thermal annealing step. The enhancement of the fluorescence lifetime of 4 I13/2 level also signifies about the crystalline environment of the surrounding Er ions induced by proper thermal annealing of pristine preform. The formation of scandium-ultra rich nano-crystalline environment, possessing low photon energy around the erbium ions, enhanced the fluorescence intensity. Such kind of nano-engineered glass reduces the noise figure around 4.35 dB and provides broadband optical flat gain with an average value of 38.675 dB, varied by less than ˙0.7 dB spanning over a broad wavelength region of 1530–1590 nm compared to the pristine and Sc free Er-doped fibers. Such kind of nano-engineered glass-based Er-doped fiber will be useful for making highly efficient optical amplifiers, suitable for present broadband optical communication systems. Yb-doped new glass composition-based nano-engineered multielements (P-ZrCe-Al-Ca)-doped optical fibers have been developed and compared the lasing as well as 633 nm absorption loss performance with standard alumino-silicate Ybdoped fiber under moderate power level. Such kind of fiber indicates that the ME-Yb-doped fiber is a promising candidate for high-power laser applications with enhanced PD resistivity. Chromium-doped phase-separated yttria-alumina-silica glass-based fiber containing Cr(IV) was fabricated by thermal annealing of the preform before fiber drawing process. The extinction coefficients in all the absorption bands corresponding to Cr3C , Cr6C , and Cr4C ions are reduced 3 times in case of fibers drawn from the annealed preform with a possible reason for this being interdiffusion of co-dopants in core and Si of cladding. The annealed preform’s fiber exhibits more intense broadband emission (500–800 nm) than the nonannealed one, which is mainly due to the presence of Cr3C ions inside the nano-structured YAS core glass. In this glass, it is possible to increase the content of Cr4C species in order to generate more wideband near-IR fluorescence at room temperature and to get a more prominent bleaching effect with a higher contrast at 1.0–1.1 m pumping. Such nano-structuration of the doping host of doped fiber through postthermal annealing of the fabricated preforms followed by fiber drawing will serve as a new route to “engineer” the local dopant environment. All the results of such kind of advanced nano-engineered glass-based optical fibers will be very much useful for making of optical fiber-based devices such as lasers, amplifiers, and sensors, which can now be realized with silica glass. In general, the composition of doped region of nano-engineered glass is crucial, and based on selected composition, it has a great potential for development of fiber laser, optical amplifier, as well as broad-band source for modern optical communication system.

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Future Work Although nano-material doping technology has opened a new way in developing novel specialty optical fibers, as far as we know, there are few work about optical fiber amplification based on silica fiber doped with nano-semiconductor materials. In future work, we will demonstrate a novel special silica fiber doped with InP and ZnS semiconductor nano-particles into the core. Due to the nano size, semiconductor nano particles will show remarkable quantum confinement effect and size tunable effect, which may provide excellent amplification features. Since this particular field is relatively in its initial stage, the future work will also target the fabrication of optical fiber comprised of suitable host of advanced nano-engineered Er/Yb and Tm/Yb co-doped modified silica glass containing 90% SiO2 through MCVD technique followed by solution doping process. The main aspect of the targeted work will reveal the enhancement of energy transfer efficiency from Yb to Er and Tm to achieve high lasing efficiency based fiber laser at NIR region through modification of the surrounding environments of RE ions embedding into different nano-engineered glass hosts.

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Fabrication of Negative Curvature Hollow Core Fiber

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Muhammad Rosdi Abu Hassan

Contents From Conventional Fibers to Photonics Crystal Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photonic Crystal Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background of Photonics Crystal Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of Hollow Core Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of Negative Curvature Hollow Core Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Importance Negative Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Guiding Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antiresonant Reflecting Optical Waveguide (ARROW) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marcatili and Schmeltzer’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coupled-Mode Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabrication of Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabrication of Negative Curvature Hollow Core Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stack and Draw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design and Properties of Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

In this chapter, we describe a review covering the development of the negative curvature hollow core fiber for the mid-IR region. The topics cover various types of hollow core fiber and their improvement made in term of attenuation of fiber, followed by a description of the guiding mechanism of the negative curvature hollow core fiber (NC-HCF) using antiresonant reflecting optical waveguide (ARROW) mechanism. Then, we present the general fabrication steps and the

M. R. Abu Hassan () Centre for Optical Fibre Technology (COFT), School of Electrical, Electronic Engineering, Nanyang Technological University, Singapore, Singapore, Singapore e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 G.-D. Peng (ed.), Handbook of Optical Fibers, https://doi.org/10.1007/978-981-10-7087-7_75

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fabrication process for negative curvature fiber. In the second part of the chapter, the design and properties of the hollow core applied in the other research work are presented. Keywords

Fiber optics · Microstructured optical fiber · Hollow core fiber · Photonics crystal fiber

From Conventional Fibers to Photonics Crystal Fibers Optical fiber is a key component of modern telecommunication technologies. This technology has been attracting much attention since the first single-mode fiber with loss T tube

T gas = T tube

reaction gases

Exhausted reaction products products Deposited glass layer

Inlet taper region

Rotating substrate tube Soot particle trajectories Traversing H2/O2 burner

Reaction initiated

Fig. 4 The deposition mechanism in MCVD

Fig. 5 Phases of soot particle evolution during deposition

thermophoresis is considered to be the main deposition mechanism by which a particle present in a temperature gradient experiences a net force in the direction of decreasing temperature (Walker et al. 1980). Further downstream, the gas and wall temperatures equilibrate causing the deposition process to stop and the nondeposited particles to flow to the exhaust tube (Park et al. 1999; Walker et al. 1980). Park et al. (1999) demonstrated that larger particles were deposited further downstream from the burner position when a high reaction temperature was used (i.e., 1700 ı C). In contrast, at low deposition temperature regime, Tang et al. (2007) demonstrated that bigger particles were deposited near the burner, while smaller ones were deposited further downstream (Fig. 5a). As the burner moves

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downstream, bigger particles will be deposited on top of the smaller ones (Fig. 5b). The movement of the burner downstream will also induce partial sintering to the deposited particles as the burner passing below the deposited soot heats the tube wall to the maximum temperature (Fig. 5c). This process is also expected to fuse several soot particles together to form bigger particles and consequently to increase the density of the soot. In this work, the same soot pattern as that observed by Tang et al. (2007) is expected to be obtained because the deposition process was conducted at a comparable temperature regime.

Soot Characteristics: Physisorption and Scanning Electron Microscope (SEM) Measurements Figure 6a, b is the isotherm curves of the excess soot collected from the exhaust tube and the deposited soot scrapped off of the substrate tube, respectively. Both of the soot samples were produced at the temperature of 1700 ı C. It should be mentioned that in our fabrication process, the burner was not allowed to pass below the exhaust tube and, therefore, the soot collected in the exhaust tube did not experience partial sintering as that deposited inside the substrate tube. The isotherm curve was obtained from 46 measurements at different pressures within 2 h 11 min of elapsed time. Each measurement was taken after the system had reached equilibrium for every change in pressure. As can be seen from Fig. 6, both adsorption isotherm curves exhibit a type IV isotherm (Sing et al. 1985). A similar shape of isotherm was also observed by Fan et al. (2008) and Rio et al. (2012). This isotherm indicates that the samples are mesoporous in nature. Typically, a mesoporous structured sample has a pore size of 5 to 20 nm (Gregg and Sing 1982; Sing et al. 1985), which induces capillary condensation of N2 at high pressure. This phenomenon is observed by the existence of a hysteresis loop near the saturation pressure po (Fig. 6a, b). The hysteresis loop occurs when the adsorption and desorption isotherms do not coincide over a certain region of external pressures (Naumov 2009). The hysteresis loop was categorized as H3 type hysteresis and is usually associated with the presence of nonrigid slit shaped pores (Naumov 2009). For samples that exhibit a type IV isotherm curves, the BET and BJH analyses have been reported to provide the most appropriate solution to determine the effective surface area and the pore volume, respectively (Gregg and Sing 1982). These soot characteristics generally indicate the amount of solution that can be impregnated inside the soot and sequentially the final dopant concentration in the preform. Dhar et al. (2008) suggested two mechanisms of the solution doping: surface adsorption and pore retention that may have a correlation with the soot surface area and pore volume, respectively. One of these mechanisms may dominate the other based on the fabrication process and parameters. Table 5 shows the effective surface area and the pore volume of the soot samples produced at different temperatures. These samples consist of soot that was scrapped off of the substrate tube and a sample of excess soot that was taken from the exhaust tube.

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Fig. 6 N2 adsorption isotherm curves for soot deposited at 1700 ı C. (a) and (b) refer to the soot collected from the exhaust tube and the substrate tube, respectively

Table 5 BET surface area and BJH pore volume of the deposited soot Substrate tube # P043 T044 P046 P023 NA P051

Temperature (ı C) 1650 1700 1750 1800 1700 (exhaust tube) 1700/1750

BET surface area(m2 g1 ) 35.1 24.5 20.9 18.0 78.7 23.2

BJH pore volume(cm3 g1 ) 0.1014 0.0640 0.0557 0.0480 0.2070 0.0523

The BET analysis shows that the excess soot has a very large effective surface area compared to the soot that was deposited on the substrate tube (cf. Table 5). This is due to the partial sintering that the soot on the substrate tube experienced during forward deposition. The additional heat that has been supplied by the forward passing burner onto the soot right after the deposition behaves as a heat treatment

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0.03 0.02 1660

1710 1760 Temperature (°C)

1810

Fig. 7 Soot BJH pore volume and BET surface area evolution with deposition temperature for single layer deposition. Solid lines are a guide for the eye

that stimulates the soot particles to fuse to one another forming bigger particles thus reducing the total surface area. This phenomenon intensifies when the soot is exposed to higher temperatures (cf. Fig. 7). For example, the soot surface area was reduced significantly from 35 m2 g1 to 18 m2 g1 when the deposition temperature was increased from 1650 to 1800 ı C (Table 5). In addition, the BET analysis of the dual-layer soot indicates that its average surface area (23.2 m2 g1 , Table 5) is close to that of the single layer soot deposited between 1700 and 1750 ı C (24.5 and 20.9 m2 g1 , respectively (Table 5)). In a dual-layer soot structure, the additional layer is deposited on top of a porous surface instead of a solid one (i.e., the tube wall). The layer upon layer structure reduces the heat transfer that, in turn, reduces the amount of sintered particles. Thus, the temperature gradient across the duallayer soot causes the average particle size of the deposited soot to be smaller than the single-layer soot for the same deposition temperature. The pore volume of the soot is another characteristic that is affected by the deposition temperature. Using BJH analysis, the pore volume is observed to decrease as the deposition temperature increases (Table 5). This is because the heat from the burner supplies enough energy to cause the pore structure to collapse and the soot to density. It is believed that there is an inversely proportional relationship between the pore volume and the density of a porous sample (Zielinski and Kettle 2013). The increment of soot density with temperature that was observed by Petit et al. (2010) is in agreement with the pore volume pattern obtained from BJH measurements carried out in this work. For the dual-layer soot structure, the pore volume is found to be comparable to the single layer soot deposited between 1750 and 1800 ı C (Table 5). The reason behind this difference in pore volume is unclear. It is suspected that due to the smaller size soot that is deposited at the bottom of

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each layer, the voids on the surface of the first layer are filled with the smaller particles that are produced by the second layer which effectively reduces the total pore volume (Tang et al. 2007). It is noteworthy to mention that a soot with a smaller pore volume is expected to be less sensitive to temperature variations and is stronger at withstanding external forces, hence improving the longitudinal uniformity. Figure 8 shows the SEM images of the soot samples deposited at temperatures ranging from 1650 to 1800 ı C using both single and dual pass deposition method. Both the top view and cross sectional side view images of the soot are displayed for each deposition temperature. As can be observed from Fig. 8a–c top view, larger soot particles were obtained at higher deposition temperatures, which is in agreement with the surface area measurements. As seen in Fig. 5b, during deposition, larger soot particles are deposited on top of smaller particles creating a gradient of increasing particle size from the tube wall to the exposed inner surface (Tang et al. 2007). However, this gradation vanishes soon after the burner passes under the deposited soot owing to the heat supplied by the burner that causes the particles to fuse to one another forming larger particles. On the other hand, at the same longitudinal position, there exists a gradient of decreasing heat distribution towards the tube axis that gives a counter effect to the aforementioned particle fusion. Tang et al. (2007) observed a greater particle fusion effect for soot nearer to the tube wall than that nearer to the exposed inner surface. In contrast, in this work, no gradation of increasing particle size is observed (Fig. 8a–c side view). This indicates that the oxy-hydrogen burner supplied enough heat to cause the particles to fuse to one another. Figure 8e shows the soot size of the dual layer P051 (top and side views). From the top view image, the soot size is considerably similar to P023 (Fig. 8c) and P046 (Fig. 8b), which is in agreement with the BET results. However, an obvious difference between the dual and single layers is observed from the side view images. The formation of the dual layer soot is apparent and the difference between both layers is noticeable. It is obvious that the top layer has higher porosity from its “fluffy” appearance. It is believed that the radial particle size distribution of the single layer deposition that was observed by Tang et al. (2007) also applies for the case of dual layer deposition. The pattern of dual layer soot is illustrated in Fig. 9 whereby the smaller size soot is located at the bottom of each layer. As compared to the single layer, it is believed that for the dual layer the gradient of increasing particle size along the radial axis decreases by half as each layer introduces individual pattern of particle size. This will produce an innermost layer (furthermost from the tube wall) with smaller particles as compared to the single layer deposition with the same total SiCl4 flow (Fig. 9). It is also believed that the average particle size will also be smaller than that of the single pass having the same total SiCl4 flow and deposition temperature. This is because the second layer is deposited on top of the first layer which is not a solid surface (i.e., a porous layer). This makes the heat transfer from the burner to the second layer becomes less effective compared to the single layer. In single layer deposition, the soot is deposited on a tube wall that has a higher thermal conductivity than a porous surface (Davarzani et al. 2011).

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Fig. 8 Top and side views of all deposited soot (Table 2): (a), (b), (c), (d), and (e) refers to 1650, 1750, 1800, 1800 (dual layer), and 1700/1750 (dual layer) ı C, respectively

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Fig. 9 Expected difference between dual- and single-layer soot patterns

Figure 8c, d compares the innermost layer (furthermost from the substrate wall) morphology of the single-layer soot to that of the dual-layer soot deposited at the same temperature (i.e., 1800 ı C). It can be observed that the particles of the single soot layer are larger and more interconnected (fused) between one another. This is due to the partial fusion effect that takes place when the burner passes the soot layer. The soot nearer to the tube wall experiences greater partial fusion since it is exposed to higher temperature. It can also be observed that the innermost soot layer of the dual layer deposition, which is further form the tube wall (due to thickness), exhibits smaller particles and lesser interconnection.

Effect of Soot Condition on the Final Preform Characteristics The characteristics of the fabricated preforms (cf. Table 2) are discussed to determine which soot deposition parameters offer the best soot performance in terms of dopant intake and longitudinal uniformity. Once the optimum soot is identified, the choice of glass modifier to be incorporated inside the preform is extended to other types of dopants besides aluminum such as gallium and barium. Figure 10 shows the longitudinal refractive index difference of the fabricated Al2 O3 doped preforms along a length of 23 cm. According to Kirchhof et al. (2003) and Vienne et al. (1996), the doping of Al2 O3 inside silica increases the refractive index by 2.3  103 per mol% of Al2 O3 . The effect of refractive index change due to rare earth (RE) oxide incorporation is negligible because of the much lower concentration of RE2 O3 in comparison to Al2 O3 . A summary of preforms characteristics is displayed in Table 6 including parameters such as average refractive index difference, nav ; maximum index difference variations, R; relative standard deviations of index difference, %RSD; and average core diameter, Dcore for each of the fabricated preforms. For the single layer soot preforms, the refractive index difference, which is related to the amount of dopant incorporated in the preform, is higher for lower deposition temperatures (Table 6). This result is in agreement with reports from several previous authors (e.g., Khopin et al. 2005; Tang et al. 2007). The higher

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0.016 0.014

Index Difference

0.012 0.010 0.008 0.006 0.004 0.002

P043

P023

P051

P046

P032

0.000 0

5

10 15 Horizontal Length (cm)

20

25

Fig. 10 Longitudinal refractive index difference of the fabricated preforms along 23 cm. Preforms P043, P046, and P023 represent the single-layer soot deposited at 1650, 1750, and 1800 ı C, respectively. Preforms P032 and P051 represent the dual-layer soot deposited at 1800 and 1700/1750 ı C, respectively

Table 6 Characteristics of the fabricated preforms Preform # P043 P046 P023 P032 P051

Tdep (ı C) 1650 1750 1800 1800 1700/1750

nav (102 ) 0.944 0.791 0.813 0.985 1.150

Average mol% 4.1 3.4 3.5 4.3 5.0

R (103 ) 0.99 0.35 0.18 0.15 0.15

RSD (%) 29.6 14.3 6.6 4.8 3.5

Dcore (mm) 1.32 1.31 1.18 1.42 1.43

dopant intake for preforms with lower soot deposition temperature is due to the higher surface area and pore volume of the soot (cf. Table 5). However, the uniformity of the preform deteriorates when a lower deposition temperature is applied. An increase of R values from 0.18 to 0.99 is observed when the deposition temperature is reduced from 1800 ı C to 1650 ı C (Table 6). This observation is in agreement with Kirchhof et al. (2003) and Petit et al. (2010) that reported an exponential reduction of dopant intake with soot density increment and a linear increase of soot density with deposition temperature increment. This means that the variation of concentration is more severe at lower deposition temperatures as compared to the higher ones due to the features that govern the soot density. Even though the measurement of soot density is not included in this work, the features that govern the soot density are expected to be the soot surface area and the pore volume because both are reduced exponentially with increased deposition temperature (cf. Table 5). This observation is in agreement with the experimental observation and the model proposed by Kirchhof et al. (2003) which was briefly explained in

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section “MCVD–Solution Doping Technique.” Hence, for a single layer deposition method, a higher incorporation of dopant is achievable but at the expense of preform longitudinal uniformity degradation. In light of the above, a dual layer soot deposition method is proposed to improve the dopant intake as well as the longitudinal uniformity of the solution doped preforms. Multiple soot layer method assisted by a presintering pass has been previously proposed by Paul et al. (2010) to fabricate a large core fiber using solution doping technique. This method, used at optimum parameters, has been found to promote uniformity of the soot porosity across the soot layer thus improving the dopant radial distribution. For this work, since the fabricated preform core is small, the radial uniformity is not of a major concern compared to the longitudinal uniformity. As shown in Table 6 and Fig. 10, the additional layer is found to improve both the dopant concentration and the longitudinal uniformity of the fabricated preforms. An improvement of 22% in index difference, 21% in dopant intake and 27% in longitudinal uniformity (%RSD) is achieved by dual layer soot deposition (P032) compared to single layer deposition (P023) fabricated at the same temperature (1800 ı C). To further increase the dopant intake, a lower process temperature is proposed for the dual layer soot deposition (P051). A lower deposition temperature of 50 ı C for the first pass compared to the second pass is also used to promote the radial uniformity. In this case, smaller particles are deposited in the first pass compared to the second pass due to the lower deposition temperature. However, this difference in particle size is eliminated during the partial sintering of the burner second pass. This is because the soot on the bottom layer experiences higher degree of particle fusion compared to the one on the top due to temperature gradient inside the tube. As shown in Table 6, a further 17% increment in both the average refractive index and dopant concentration is achieved by lowering and varying the deposition temperatures of the dual layer soot preform. However, the average n fluctuation, R, is similar to that of P032 (i.e., 0.15  103 ) even though the %RSD value improves by 27%. An R value of 0.15  103 corresponds to a 0.02 fluctuation in the numerical aperture, NA, of the final fiber along 23 cm of the preform. The improvement of P051 characteristics as compared to the single layer soot preform is attributed to the higher soot surface area and pore volume as aforementioned (cf., Table 5). The average core diameter of all the fabricated preforms is also presented in Table 6. Since two layers of soot are deposited, the average Dcore of P032 and P051 is larger as compared to the rest of the single-layer soot preforms. It is also observed that for the same number of soot layers, the core size increases with Al2 O3 content. The evolution of Dcore along the preform length is depicted in Fig. 11. It can be observed that the Dcore pattern closely emulates the longitudinal n pattern along the length of the preform. The concentration of Al dopant is also determined using energy-dispersive X-ray spectroscopy (EDX) point ID measurements. Figure 12 displays an example of EDX analyses carried out for P051. As can be noticed from the figure, the Al distribution profile across the preform’s core closely resembles the refractive index profile (RIP). It should be mentioned that thulium is not detected by this method since its concentration falls below the EDX detection limit.

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1.8 1.7 Core Diameter (mm)

1.6 1.5 1.4 1.3 1.2 1.1 P046 P023 P051

1 0.9 0.8

0

5

10 15 Preform Length (cm)

P043 P032

20

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Fig. 11 Core diameter variation along the fabricated preforms

6

Al wt%

Al2O3 (mol%)

0.014

Index difference

0.012 0.01

4 0.008 3 0.006 2

Index Difference

Concentration (%)

5

0.004

1

0.002

0 -1

-0.6

-0.2

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Radial Position (mm) Fig. 12 EDX measurements of Al concentration across the core of P051. Inset is the EDX spectrum

Alumina, Gallia, and Baria Solution Doped Silica Preforms Table 7 lists a summary of the preforms fabricated in this work. The characteristics of the individual preforms are given in sections “Aluminum Doped Preforms,” “Gallium Doped Preforms,” and “Barium Doped Preforms.”

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Table 7 Summary of the fabricated Ga, Ba, and Al preforms Preform # P053 P057 P085 P058 P060

Modifier type Al Ba Ba/Al Ga Ga

Modifier(mol%) 6.1-8.3 3.0 1.2/1.6 2.7 1.7

nmean (103 ) 15.4 10.0 5.5 8.8 5.6

%RSD 13.5 7.3 5.7 9.5 6.3

Soot type Dual-layer Dual-layer Dual-layer Dual-layer Dual-layer

Fig. 13 Longitudinal n along P053 length. Inset is the RIP at 17 cm from the preform inlet

Fig. 14 Longitudinal image of P053

Aluminum Doped Preforms The RIP of Al doped P053 is presented in Fig. 13. Using a 3.2 M Al solution, a maximum n value of 0.019, which corresponds to 8.3 mol%, is achieved. However, the longitudinal n fluctuates from 0.014 to 0.019 with %RSD of 13.4%. The lower end of n distribution corresponds to about 6.1 mol% of Al2 O3 . As can be observed from Fig. 14, some parts of the preform are phase separated due to the high aluminum concentration. The low uniformity of the preform may be attributed to the high solution strength that is used to impregnate the soot layer. Preforms that are soaked using high concentration solutions are more sensitive to the difference in density of the deposited soot layer as described above.

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0.01 0.009

Index Difference

0.008 0.007 0.006 P058 P060

0.005 0.004 0.003 0.002 0.001 0 -10

-5

0

5

10

Radial Distance (mm) Fig. 15 Refractive index profiles of P058 and P060

Gallium Doped Preforms Figure 15 displays the refractive index profiles of the fabricated gallium doped preforms P058 and P060. In addition, Fig. 16 illustrates the longitudinal refractive index profiles along the length of P058 and P060. The higher index difference of P058 is attributed to the higher concentration of Ga(NO3 )3 .H2 O solution used in soaking the porous layer of the preform (i.e., 2.3 M, Table 3). However, the gallium doped preforms exhibit a large dip at the center of the preforms’ core. The index dip is a consequence of the evaporation of the dopant material inside the core. It is known that the evaporation of the dopant becomes more rapid during the collapse stage where the tube is at its highest temperature (Cognolato 1995). From Fig. 15, the collapse process for P058 and P060 is found to result in the loss of 45% of Ga2 O3 from the center of the preforms’ core. Figure 17a, b shows the side view of P058 and P060, respectively. As can be seen from Figure 17a, a slight a slight opalescence appears in the center of P058 between the lengths of 1 to 10 cm measured from the preform input end. The occurrence of opalescence indicates that the P058 preform core is phase separated. Phase separation only occurs for glass systems of multiple compositions (Shelby 2005). It is common for a multicomponent glass to be immiscible (i.e., a homogeneous mixture is thermodynamically unfavorable) at a certain composition concentration and temperature. The EDX measurements shown in Fig. 18 are carried out for samples taken from the opalescent region of P058. It can be deduced from the figure that the phase separation occurs when  2.7 mol% of Ga2 O3 is doped inside the preform core. This shows that gallium has a lower solubility in silica as compared to aluminum. However, no sign of opalescence is observed for P060 (Fig. 17b), which indicates that phase separation is absent at the respective Ga2 O3 concentration

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P060 P058

0.01 Index Difference

575

0.008 0.006 0.004 0.002 0

0

5

10 Preform Length (cm)

15

20

Fig. 16 Longitudinal refractive index difference along the length of P058 and P060

Fig. 17 a, b Longitudinal view of P058 and P060, respectively

(i.e.,  1.7 mol%). Schneider and Syms (1998) reported a gallium silica glass system having a 10% P2 O5 that was fabricated using sol gel technique. They observed a clear glass (miscible glass system) as long as the Ga2 O3 concentration was below 2.5 mol%. The EDX spectrum of P058 is depicted in Fig. 19. There are 5 elements detected in the core (i.e., Si, Ga, O, C, and Cl). The peaks at 1.739, 9.241, and 0.525 keV are assigned to Si, Ga, and O, respectively. The small peaks at 0.277 and 2.621 keV are attributed to C and Cl, respectively. The existence of C element is due to the coating layer applied to the sample to avoid the accumulation of charges on the nonconducting surface of the measured sample. The presence of Cl may be attributed to residual chlorosiloxanes (i.e., intermediate compounds of the SiCl4 oxidation reaction) that may have been trapped inside the silica matrix.

Barium Doped Preforms The refractive index profiles of barium doped preforms P057 and P085 are shown in Fig. 20. P057 has a higher n than P085 due to the higher concentration of BaCl2 .H2 O solution used in the solution doping process (1.4 vs. 0.9 M, respectively,

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7

0.01 0.009

6

0.007

4

0.006 0.005

3

0.004

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Ga (wt%) Ga2O3 (mol%) Index difference

1 0 0

0.4

Index difference

Concentration (%)

0.008 5

0.001 0

0.8 1.2 Radial position (mm)

1.6

Fig. 18 EDX concentrations of gallium and gallium oxide (primary axis); and the corresponding refractive index difference (secondary axis) for P058 Fig. 19 Energy-dispersive X-ray (EDX) spectrum of P058

O Kα Si Kα C Kα

Ga Kα Cl Kα

Table 3). The dip in the center of the RIP of P057 is not as large as that observed for the gallium preforms (cf. Fig. 15) indicating lesser BaO evaporation during the collapse process. However, the RIP of P085 exhibits a distortion in the center of the core, which is indicative of crystal formation during the fabrication process. It is also worth noting that the core diameter of P057 is 57% higher than that of P085. This may be attributed to the higher barium concentration and the occurrence of phase separated core in P057 (Sudo 1997). The longitudinal images of P057 and P085 are depicted in Fig. 21a, b, respectively. As observed in this figure, P057 possesses a slight bluish-milky white opalescent core that is indicative of the presence of phase separation. This is in agreement with the observation made by Seward et al. (1968). Utilizing normal and fast quenching techniques, Seward et al. obtained the liquid-liquid immiscibility gap in the range of 4 to 28 mol% of BaO. Even though a clear glass was observed at

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0.012

Index Difference

0.01 0.008 P085

0.006

P057

0.004 0.002 0 -10

-5

0 Radial Distance (mm)

5

10

Fig. 20 Refractive index profiles of P057 and P085

Fig. 21 Images of P057 (a) and P085 (b)

2 mol% of BaO, barium-rich particles sized between 20 to 60 nm were observable in the system indicating the occurrence of phase separation. For the MCVD coupled with solution doping technique used in this work, the opalescence is observed at  3 mol% of BaO. The opalescence at such low BaO concentration may be attributed to the slower cooling experienced by the glass due to the slow moving burner which gives ample time for nucleation and growth to take place. The phase separation phenomenon in BaO-SiO2 system can be suppressed by either fast cooling from the temperature above the immiscible gap or by adding Al2 O3 to the system. The former technique can be conducted during fiber pulling, while the latter one motivated the fabrication of P085 whereby Al2 O3 was added to the preform by using a solution containing both Ba and Al ions (cf. Table 3). The concentration of the dopant precursors was chosen so that the final concentration (mol%) for both oxides inside the preform would be similar and that the solution did not reach its saturation point. As observed in Fig. 21b, the addition of aluminum to the BaO-SiO2 system successfully suppresses the phase separation and the presence of opalescent core is not observed along the entire preform length.

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Index Difference (cm)

0.012 0.01 0.008 0.006 0.004 P057

0.002 0 0

P085

5

10 15 Preform Length (cm)

20

Fig. 22 Longitudinal refractive index difference along P057 and P085 length

The longitudinal refractive index difference of P057 and P085 along the preform length is shown in Fig. 22 with average n values of 0.010 and 0.0055, respectively. The longitudinal uniformity of P085 is found to be better than P057 by 14%. This is due to the use of higher solution concentration in P057 as compared to P085. As discussed in section “MCVD–Solution Doping Technique,” high concentration solutions impose higher fluctuation of dopant concentration for a small change of soot density along the preform as compared to weaker ones. The refractive index difference and its corresponding oxide concentration for P057 and P085 are depicted in Fig. 23a, b, respectively. The EDX spectrum for P057 is shown in Fig. 24. The inset in the figure shows the spectrum obtained for P085. Both of these preforms exhibit four common peaks at 1.739, 4.46, 4.82, and 0.525 keV corresponding to Si, Ba, Ba, and O, respectively. The bombarded electrons induce two characteristic X-ray emissions that are assigned to barium (i.e., 4.46 and 4.82 keV). For P085, there are two extra peaks at 0.277 and 1.487 keV corresponding to carbon and aluminum, respectively. The carbon peak is believed to originate from the carbon coated layer that is used to prevent charging of the glass by the electron beam.

Spectroscopic Characteristics of Thulium Doped Fibers (TDF) After the fabrication process of the optical fiber, which consists of preform making and fiber pulling, is completed, the thulium ions spectroscopy in the modified silica fiber is analyzed. The effect of modifiers on the absorption spectrum and the fluorescence decay lifetime of thulium ions is also observed and analyzed. These two measurements provide information on how feasible the modification of the silica structure improves the overall characteristics of the TDF.

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0.012

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0.008

4 0.006 3 0.004

2 Ba wt% BaO Index Difference

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1.6 Concentration (mol%)

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Index Difference

Concentration (%)

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579

1.4 1.2 1 0.8 0.6 0.4

BaO

0.2

Al2O3

0 0

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1 Radial position (mm)

1.5

Fig. 23 (a) EDX concentration of barium and barium oxide (primary axis); and the corresponding index difference (secondary axis) of P057. (b) EDX concentration of dopant modifiers inside P085

Absorption Figure 25 shows the normalized absorption spectra for a number of fabricated Tm doped fibers. The range of the measured spectra is from 600 to 1750 nm. It can be observed that within this range, there exist four main absorption peaks that correspond to the ground state absorption of thulium ions. These peaks represent the energy manifolds of thulium ions listed in Table 8. The thulium concentration can be determined from the absorption spectrum of the fiber (Hanna et al. 1990) and is also listed in Table 8 for the tested fibers. As observed from this table, the amount of Tm ions doped in the fibers is dependent on the type of modifier that is co-doped in the glass matrix. By using the same strength of TmCl3 inside the solution (i.e., 0.025 M), the highest concentration of Tm ions is obtained by Ba/Al

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Fig. 24 Energy-dispersive X-ray (EDX) spectrum for P057. The inset is the EDX spectrum obtained for P085 Fig. 25 Normalized absorption spectra of various Tm doped fibers. Tmdf200 is the commercially available Tm doped fiber obtained from OFS/Furukawa Company

doping (8791 ppm) followed by Al (3408 ppm), Ga (2383 ppm), and Ba (629 ppm) (Table 8). This indicates the low solubility of rare earth ions inside BaO-SiO2 glass matrix.

Lifetime The values of Tm ions lifetime for the tested fibers are listed in Table 9. The decay lifetime is the time for the intensity to drop to 1/e from its maximum value. This is accurate for a single exponential decay whereby the lifetime value is taken as the decay constant, £ of the decay waveform (i.e., I(t) D Io et/ ). However, for multisites

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Table 8 Thulium energy manifold peak absorption and the respective estimated concentration 3

F023 F057 F058 F085 Tmdf200

F4 ’p (dB/m) 29.9 7.0 24.0 76.2 30.4

3

H5 ’p (dB/m) 50.9 19.7 39.5 130.2 64.8

3

H4 ’p (dB/m) 81.8 15.1 57.2 211.0 79.5

3

F2 , 3 F3 ’p (dB/m) 30.7 7.29 22.6 59.6 37.8

Concentration (ppm) 3408 629 2383 8791 3312

Table 9 Measured lifetimes of various Tm doped fibers Fiber# F023 F057 F058 F085 Tmdf200 Pure silica1 1

Modifier type Al Ba Ga Ba/Al n.a –

Modifier concentration(mol%) 3.5 3.0 2.7 1.2/1.6 n.a. –

Lifetime (s)3 H4 27 28 30 23 33 12.4

Lifetime (s)3 F4 490 477 561 360 572 327

Simpson (2008)

luminescence of active centres, where there exist multiple decay constants, a new method is needed to better represent the fluorescence decay curve. Aronson (2006) proposed the use of the time constant of the single exponential curve fit as the lifetime value. Nevertheless, for this work, the lifetime value is determined using the conventional technique. The longest 3 H4 and 3 F4 lifetimes of the tested fibers are obtained from Tmdf200 fiber (33 and 572 s, respectively) followed by F058 (30 and 561 s, respectively). Although the concentration of Ga is quite low compared to alumina (2.7 vs. 3.5 mol%, respectively), the obtained 3 H4 lifetime is longer (30 vs. 27 s, respectively). This is due to Ga being heavier than Al which in turn lowers the nonradiative decay. However, the setback of Ga is that it can only be doped at low concentrations due to its lower solubility in silica as compared to alumina. Hence, the higher solubility of Al in silica (up to 9 mol%) serves it as a better candidate than Ga. Faure et al. (2007) fabricated an 8 mol% aluminosilicate fiber that exhibited a 3 H4 lifetime of 54 s. In addition to Al and Ga, the 3 H4 lifetime of 3 mol% Ba TDF (i.e., F057) is found to be 28 s (Table 9), which falls between that of Ga and Al with comparable dopant concentration. The shortest lifetime for both 3 H4 and 3 F4 transitions is demonstrated by F085 (23 and 360 s, respectively). This is due to the low dopant (Al and Ba) concentration inside the fiber. Hence, the nonradiative decay is higher in F085 as compared to fibers with higher dopant concentration and the lifetime is consequently shorter. In addition, the high concentration of Tm ions in F085 as compared to other

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Fig. 26 Decay spectra of 3 H4 energy level for various Tm doped fibers

Fig. 27 Decay spectra of 3 F4 energy level for various Tm doped fibers

fibers may increase the cross relaxation and co-operative up-conversion rates which consequently decrease the measured fluorescence lifetime. Figures 26 and 27 provide a better visualization for the 3 H4 and 3 F4 decay lifetimes of the tested fibers, respectively. It can be observed that the decay lifetime of 3 F4 closely resembles that of 3 H4 for the tested fibers. This can be explained by the high phonon energy of the glass that consistently influences the nonradiative decay rates for each Tm energy level.

Conclusions The main objective of this work is to improve the Tm ions emission in silica fibers. This is done by doping the TDF with modifier metals such as Al, Ba, and Ga with the aim to reduce the high phonon energy environment surrounding Tm ions in pure silica system. The fibers were fabricated using an in-house MCVD-solution doping system. In order to maximize the concentration of the proposed modifier ions inside the glass preform, the solution doping method was optimized and improved.

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A deposition temperature in the range of 1700 to 1800 ı C was found to be effective in producing a fairly high dopant intake and an acceptable longitudinal uniformity. The dopant intake and the longitudinal uniformity of the preforms were then further improved by introducing an additional layer of soot (i.e., dual-layer soot) deposited at 1800 ı C. This dual-layer structure is believed to have a different morphology as compared to the single layer one which in turn increased the dopant intake as well as the longitudinal uniformity of the preform by 21% and 27%, respectively. Further improvement was also obtained by using deposition temperatures of 1700 and 1750 ı C for the first and second soot layers, respectively. This resulted in a further 17% increase in dopant intake. This type of dual-layer soot was then used to fabricate all the Al, Ba, and Ga doped preforms. Using this improved soot, three preforms doped individually with Ga, Ba, and Al were fabricated using the highest solution concentration. All of the fabricated preforms exhibited opalescence core indicating that the amount of the modifier oxide had reached its limit inside silica. The maximum concentrations of Ga, Ba, and Al attained inside these preforms are  2.7, 3.0, and 8.3 mol%, respectively. Compared with pure silica host, lifetime measurements showed that the lifetime of the 3 H4 manifold increased by  218%, 226%, and 242% when doping with alumina, baria, and gallia, respectively. A similar improvement was also obtained for the 3 F4 manifold radiative transition with increments of 150%, 146%, and 172%, respectively. This showed significant improvement in obtaining radiative emission using the modifiers proposed in this work.

References A.N. Abramov, M.V. Yashkov, A.N. Guryanov, M.A. Melkumov, D.A. Dvoretskii, I.A. Bufetov, L.D. Iskhakova, V.V. Koltashev, M.N. Kachenyuk, M.F. Torsunov, Inorg. Mater. 50, 1283 (2014) B.G. Aitken, M.L. Powley, R.M. Morena, B.Z. Hanson, J. Non-Cryst. Solids 352, 488 (2006) K. Arai, H. Namikawa, Y. Ishii, H. Imai, H. Hosono, Y. Abe, J. Non-Cryst. Solids 95, 609 (1987) J. E. Aronson, Ph.D. thesis, University of Southampton, 2006 R. M. Atkins, R. S. Windeler, U. S. Patent No. US20030167800A1 (11 Sept 2003) M. Binnewies, K. Jug, Eur. J. Inorg. Chem. 2000, 1127 (2000) W. Blanc, T.L. Sebastian, B. Dussardier, C. Michel, B. Faure, M. Ude, G. Monnom, J. Non-Cryst. Solids 354, 435 (2008) S. Brunauer, P.H. Emmett, E. Teller, J. Am. Chem. Soc. 60, 309 (1938) L. Cognolato, J. Phys. IV 5, 975 (1995) B. J. Cole, M. L. Dennis, U. S. Patent No. US6667257B2 23 Dec 2003 H. Davarzani, M. Marcoux, M. Quintard, Int. J. Therm. Sci. 50, 2328 (2011) M. Dennis, B. Cole, S-band amplification in a thulium doped silicate fiber, in Optical Fiber Communication Conference and International Conference on Quantum Information, 2001. OSA Technical Digest Series (Optical Society of America, 2001), Anaheim, California, paper TuQ3. https://doi.org/10.1364/OFC.2001.TuQ3 A. Dhar, M.C. Paul, M. Pal, A.K. Mondal, S. Sen, H.S. Maiti, R. Sen, Opt. Express 14, 9006 (2006) A. Dhar, A. Pal, M.C. Paul, P. Ray, H.S. Maiti, R. Sen, Opt. Express 16, 12835 (2008) E.M. Dianov, J. Lightw. Technol. 31, 681 (2013) W. Fan, M.A. Snyder, S. Kumar, P.-S. Lee, W.C. Yoo, A.V. McCormick, R.L. Penn, A. Stein, M. Tsapatsis, Nat. Mater. 7, 984 (2008)

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B. Faure, W. Blanc, B. Dussardier, G. Monnom, P. Peterka, Thulium-doped silica-fiber based S-band amplifier with increased efficiency by aluminum co-doping, in Optical Amplifiers and Their Applications/Integrated Photonics Research, Technical Digest (CD) (Optical Society of America, 2004), Francisco, California, paper OWC2. https://doi.org/10.1364/OAA.2004.OWC2 B. Faure, W. Blanc, B. Dussardier, G. Monnom, J. Non-Cryst. Solids 353, 2767 (2007) G.N. Greaves, J. Non-Cryst. Solids 71, 203 (1985) S.J. Gregg, K.S.W. Sing, Adsorption, Surface Area and Porosity (Academic Press, London, 1982) A. Halder, M.C. Paul, S.W. Harun, S.K. Bhadra, S. Bysakh, S. Das, M. Pal, J. Lumin. 143, 393 (2013) A. Halder, M.C. Paul, S.K. Bhadra, S. Bysakh, S. Das, M. Pal, Sci. Adv. Mater. 7, 631 (2015) D.C. Hanna, I.R. Perry, J.R. Lincoln, J.E. Townsend, Opt. Commun. 80, 52 (1990) D.W. Hewak, R.S. Deol, J. Wang, G. Wylangowski, J.A.M. Neto, B.N. Samson, R.I. Laming, W.S. Brocklesby, D.N. Payne, A. Jha, M. Poulain, S. Otero, S. Surinach, M.D. Baro, Electron. Lett. 29, 237 (1993) P. Jander, W.S. Brocklesby, IEEE J. Quantum Electron. 40, 509 (2004) V.F. Khopin, A.A. Umnikov, A.N. Gur’yanov, M.M. Bubnov, A.K. Senatorov, E.M. Dianov, Inorg. Mater. 41, 303 (2005) J. Kirchhof, A. Funke, Cryst. Res. Technol. 21, 763 (1986) J. Kirchhof, S. Unger, A. Schwuchow, Fiber lasers: materials, structures and technologies, in Proc. SPIE4957, Optical Fibers and Sensors for Medical Applications III (2003), pp. 1–15 C.J. Koester, E. Snitzer, Appl. Opt. 3, 1182 (1964) S. Naumov, Hysteresis Phenomena in Mesoporous Materials (Universität Leipzig, Leipzig, 2009) S. Ohara, N. Sugimoto, Y. Kondo, K. Ochiai, Y. Kuroiwa, Y. Fukasawa, T. Hirose, H. Hayashi, S. Tanabe, Bi2 O3 -based glass for S-band amplification, in Proceedings of SPIE4645, Rare-EarthDoped Materials and Devices VI (2002), pp. 8–15 K.S. Park, B.W. Lee, M. Choi, Aerosol Sci. Technol. 31, 258 (1999) M.C. Paul, B.N. Upadhyaya, S. Das, A. Dhar, M. Pal, S. Kher, K. Dasgupta, S.K. Bhadra, R. Sen, Opt. Commun. 283, 1039 (2010) D.N. Payne, Electron. Lett. 23, 1026 (1987) V. Petit, A. Le Rouge, F. Béclin, H. El Hamzaoui, L. Bigot, Aerosol Sci. Technol. 44, 388 (2010) S. B. Poole, Fabrication of Al2 O3 co-doped optical fibres by a solution-doping technique, in Fourteenth European Conference on Optical Communication (1988) (ECOC 88), pp. 433–436 D. Río, A. Aguilera-Alvarado, I. Cano-Aguilera, M. Martínez-Rosales, S. Holmes, Mater. Sci. Appl. 3, 485 (2012) V.M. Schneider, R.R.A. Syms, Electron. Lett. 34, 1849 (1998) K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, H. Bartelt, Adv. Opt. Technol. 3, 447 (2014) R. Sen, A. Dhar, M. C. Paul, H. S. Maiti, Patent No. WO2010109494A2, (30 Sept 2010). T.P. Seward, D.R. Uhlmann, D. Turnbull, J. Am. Ceram. Soc. 51, 278 (1968) J.E. Shelby, Introduction to Glass Science and Technology (Royal Society of Chemistry, Cambridge, 2005) P.G. Simpkins, S. Greenberg-Kosinski, J.B. MacChesney, J. Appl. Phys. 50, 5676 (1979) D. A. Simpson, Ph.D. Thesis, Victoria University, 2008 K.S.W. Sing, D.H. Everett, R.A.W. Haul, L. Moscou, R.A. Pierotti, J. Rouquérol, T. Siemieniewska, Pure Appl. Chem. 57, 603 (1985) E. Snitzer, J. Appl. Phys. 32, 36 (1961) J. Stone, C.A. Burrus, Appl. Phys. Lett. 23, 388 (1973) S. Sudo (ed.), Optical Fiber Amplifiers: Materials, Devices, and Applications (Artech House, Boston, 1997) F.Z. Tang, P. McNamara, G.W. Barton, S.P. Ringer, J. Non-Cryst. Solids 352, 3799 (2006) F.Z. Tang, P. McNamara, G.W. Barton, S.P. Ringer, J. Am. Ceram. Soc. 90, 23 (2007) F.Z. Tang, P. McNamara, G.W. Barton, S.P. Ringer, J. Non-Cryst. Solids 354, 1582 (2008) J.E. Townsend, S.B. Poole, D.N. Payne, Electron. Lett. 23, 329 (1987)

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M. Vermillac, H. Fneich, J.-F. Lupi, J.-B. Tissot, C. Kucera, P. Vennéguès, A. Mehdi, D.R. Neuville, J. Ballato, W. Blanc, Opt. Mater. 68, 24 (2017) G.G. Vienne, W.S. Brocklesby, R.S. Brown, Z.J. Chen, J.D. Minelly, J.E. Roman, D.N. Payne, Opt. Fiber Technol. 2, 387 (1996) K.L. Walker, F.T. Geyling, S.R. Nagel, J. Am. Ceram. Soc. 63, 552 (1980) B. Walsh, N. Barnes, Appl. Phys. B Lasers Opt. 78, 325 (2004) D.L. Wood, J.B. Macchesney, J.P. Luongo, J. Mater. Sci. 13, 1761 (1978) J. M. Zielinski, L. Kettle, Physical characterization: surface area and porosity (Intertek Chemicals and Pharmaceuticals (2013). www.intertek.com/chemicals. Accessed 11 Jan 2018

Microfiber: Physics and Fabrication

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Horng Sheng Lin and Zulfadzli Yusoff

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wave Equation for Microfiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adiabaticity Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabrication Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabrications of Meso Taper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabrications of Short Taper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabrications of Long Taper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application in Structural Health Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microfiber-Based IMZI Sensor Packaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microfiber-Based IMZI Sensor Deployment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

588 588 588 594 597 597 604 607 608 609 613 618 618

Abstract

In this chapter, several essential concepts for the understanding of microfiber are provided. The effective refractive indices of core mode and cladding modes corresponding to various diameters are comprehensively explained based on wave equation. Subsequently, the effective refractive indices are related to the adiabaticity criterion of microfiber based on the upper boundary of taper angle.

H. S. Lin Universiti Tunku Abdul Rahman, Sungai Long Campus, Kajang, Malaysia e-mail: [email protected] Z. Yusoff () Multimedia University, Persiaran Multimedia, Cyberjaya, Malaysia e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 G.-D. Peng (ed.), Handbook of Optical Fibers, https://doi.org/10.1007/978-981-10-7087-7_74

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Following that, an overview of microfiber fabrication techniques is reviewed. Eventually, the most recent deployment of microfiber sensor for structural health monitoring application is demonstrated.

Introduction A century ago, a fiber with an estimated diameter of 2 m, which was known as quartz thread, was drawn by Richard Threlfall (1898). It was fabricated for mechanical applications but not for light transmissions. The application of optical fiber in the fiber-optic communication industry was proposed by the Nobel laureate, Charles Kao, and his colleague George Hockham after discovering the possibility of low-loss lightwave guiding with high-purity glasses in 1966 (Kao and Hockham 1966). Since then, standard optical fibers with a diameter of 125 m have been installed worldwide in the fiber-optic communication system. The continuous research has been executed to alter the physical and optical properties of the optical fiber. It can be achieved by manipulating the geometrical and refractive index profiles of the optical fiber. The most significant work to alter the geometrical property of optical fiber is by tapering down the diameter of optical fiber to a microfiber. The microfiber was first reported in the 1980s (Burns et al. 1985, 1986; Love and Henry 1986; Love 1987). In the following decades, it was comprehensively studied on its adiabaticity and the fabrication approaches (Love et al. 1991; Black et al. 1991; Birks and Li 1992). The researchers found its possible application as optical couplers (Takeuchi and Noda 1992; Bilodeau et al. 1987; Dimmick et al. 1999), filters (Lacroix et al. 1986; Alegria et al. 1998), and sensors (Bobb et al. 1990) in the same decades. In this chapter, the microfiber adiabaticity delineation criterion is deduced based on wave equation. Subsequently, after the fabrication, the microfiber adiabaticity is justified. The following section classifies the microfiber fabrication techniques into three categories based on its waist length, illustrating each technique through examples in between the subsections. In the final section, in conjunction to the emergence of industrial revolution 4.0, the microfiber-based inline Mach-Zehnder interferometric (IMZI) sensor is designed with the consideration to cater Internet of things (IoT) integration. The deployment of microfiber-based IMZI sensor is demonstrated starting from the work of sensor fabrication to the field test sensing result on concrete structural beam.

Principle Wave Equation for Microfiber A microfiber is also known as fiber taper. It has the varying fiber diameter with z-dependent refractive index profile of n.x; y; z/ which can be illustrated as Fig. 1a. The local-mode fields are approximately constructed by modeling the fiber as a

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Fig. 1 (a) A nonuniform diameter varies along axial direction (z-axis) with refractive index profile of n.x; y; z/. (b) An approximation model with a series of cylindrical sections in the length of ız

series of cylindrical sections as shown in Fig. 1b. Hence, it is important to solve the effective refractive indices of the corresponding modes so that propagation local modes for each section are determined. Principally, a two-layer step-index fiber model is employed to simulate light propagation in optical fiber, and the layers are core and cladding. Light field is well confined in the core with surrounding evanescent field that is bounded within the cladding. In other words, light field does not reach external surface of the cladding in two-layer step-index fiber model as shown in Fig. 2. However, this model is not valid for microfiber consisting of the untapered ends, taper, and tapered waist. The light field expands in the taper region, and it is no longer confined close to the core. A three-layer step-index fiber model is required to simulate light interaction with the third layer, which is the surrounding layer such as air, coating, etc. Furthermore, in the tapered waist region, the core is so small in diameter that it can be neglected, and the two-layer step-index fiber model can again be used for cladding-surrounding. In a SMF, the fundamental mode, also known as LP mode, LP01 is the only propagation mode in the fiber. The propagation mode is known as a core mode if its effective refractive index is bounded within the condition of neff > nco , where nco is the refractive index of core. The weakly-guiding approximation is applied to identify the refractive index of the core mode. The approximation is used due to only small deviation between the core and cladding refractive indices. By solving a set of scalar wave equations with the continuity of the solution and its first derivative at the core-cladding interface, the propagation constant of each local LP modes along the fiber taper is obtained (Tsao et al. 1989). In the propagation of electromagnetic waves along z-direction, the axial axis of fiber must satisfy the scalar wave equation in cylindrical coordinate .r; ; z/ as

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1.0195

a

0.8921

1.100 0.825

0.7646

0.550 0.275

1.100 0.6372

0.00 29.6

0.825 0.550

22.2

0.5098

0.275 0.000 14.6

14.8

Y

0.3823

7.2 7.4

0.2549

–0.2

0.0

0.1275

X

–7.6 –15.0

0.0000

b 1.12

1

Arbitrary Unit

0.8

0.6

0.4

0.2

0 –10

0

10

Y Diameter (µm)

Fig. 2 (a) Light field is confined close to the diameter of the core with infinite-cladding geometry. (b) Cross section of the beam profile

14 Microfiber: Physics and Fabrication



591

1 @2 @2 1 @ C C C k 2 n2  ˇ 2 @r 2 r 2 @r r 2 @ 2

 D0

(1)

where is the electric or magnetic field, n is the refractive index profile, the wave number k D 2= in terms of free space wavelength , and the propagation constant ˇ D k neff . The core mode exists in an infinite-cladding geometry, and its neff is bounded in the condition of ncl < neff < nco , where ncl is the refractive index of cladding. The thickness of the cladding is assumed to be infinite, and the cladding-air interface is negligible. The modal parameters are defined as q u1 D k n2co  n2eff

(2a)

q w D k n2eff  n2cl

(2b)

and the solutions of Eq. 1 are given as ( D

AJ .u1 r/e iv ; CK .wr/e

iv

;

if 0 < r < co I if co < r < cl :

(3)

where A and C are constants and co and cl are the core and cladding radii, respectively. J and K are the th-order Bessel function of the first kind and modified Bessel function of the second kind, respectively. A and C are related with the continuity of electromagnetic field and the first derivative @@r at the core boundary r D co . This leads to a set of eigenvalue equations, and the mode condition is stated as u1

JC1 .u1 co / KC1 .wco / Dw J .u1 co / K .wco /

(4)

The eigenvalues solved by Eq. (4) are known as ˇm where  is the order of the Bessel function and m is the root number of Eq. (4). Similarly, the subscript m is indicated in the LP mode as LPm . Cladding mode with effective index of neff is guided within the cladding bounded with core and air medium which satisfies the condition of nair < neff < ncl , where nair is the refractive index of the air. Hence, a three-layer step-index fiber model is applied to evaluate the light field on two boundary conditions, which are corecladding and cladding-air interfaces (Erdogan 1997). The modal parameters are defined as q u1 D k n2co  n2eff (5a) q u2 D k n2cl  n2eff

(5b)

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q w3 D k n2eff  n2air

(5c)

Therefore, Eq. (1) is solved as 8 i ˆ ˆ cl :

where A, B, C , and D are just constants while Y is the th-order of Bessel function of second kind. Similarly, the A, B, C , and D constants are related to the continuity of electromagnetic field and the first derivative @@r at the core boundary r D co and cladding boundary r D cl . This leads to two sets of eigenvalue equations, and the mode conditions are stated in the matrix form at core boundary, r D co 14) enables the propagations of higherorder modes. This requires careful conditions for launching THz waves into the hollow-core to excite the HE11 mode only, which has the lowest attenuation coefficient and the fastest velocity. The propagation of multimodes has also an

26 Optical Fibers in Terahertz Domain

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Fig. 14 (a) Attenuation spectra of the HE11 mode propagated within the hollow-core of a PS-/Aucoated hollow-core THz fiber, obtained from numerical calculations with a mode solver based on the finite element method (FEM) or from Eq. (8). The 2D distribution of the Poynting vector of the HE11 mode at 1.71 and 3.65 THz (with material absorption of the PS layer) is shown in the insets. (b) Evolution of the effective index and group velocity dispersion of the HE11 mode versus frequency. These curves are obtained from numerical calculations with a mode solver based on FEM. The hollow-core of a PS-/Au-coated hollow-core THz fiber is composed of a core diameter of D D 2 mm, a PS layer of t D 25 m with a refractive index of nd D 1.58 – i*3.58103 , and a gold layer with refractive index of n D 356 – i*444. For simplicity, these values are considered constant of this frequency range. FEM-based calculations are realized with or without the material absorption of the PS layer (i.e., imaginary part of nd )

impact to the propagation of pulses by deteriorating and elongating the waveform (in time domain) due to different mode velocities. As shown in Fig. 16, the 2D electric field (Ex) distribution of different modes, at the fiber output, could be selectively measured by varying the launching conditions of THz waves into the

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Fig. 15 (a) Attenuation coefficient measured with the cutback method for PS-/Ag-coated hollowcore THz fibers with a core diameter of 1.7 or 2.2 mm and a PS layer with different thickness. (b, d) Schematic illustrations of the electric field lines of (b) the TE01 mode and (d) the HE11 mode. (c) Measured 2D intensity distribution of THz waves (at 2.5 THz) propagated through 45cm-long length of a PS-/Ag-coated hollow-core fiber composed of a core diameter of 1.6 mm and a PS layer thickness of 2 m. Measured 2D intensity distribution of THz waves (at 2.5 THz) propagated through 90-cm-long length of a PS-/Ag-coated hollow-core fiber composed of a core diameter of 1.6 mm and a PS layer thickness of 10 m. (Reproduced from Bowden et al. (2008), with the permission of AIP Publishing)

Fig. 16 (a) Measured spatial 2D distribution of the normalized electric field (Ex) of the first modes (HE11 , TE01 , HE12 , and HE13 , respectively) propagated through an PS-/Ag-coated hollow-core THz fiber (Dcore D 1.8 mm, t D 14 m) over a fiber length of 133.5 mm. The measures of different modes have been realized by varying the launching conditions of THz radiations into the hollowcore and by extracting the map (2  2 mm2 area) at different times (t D 0, 1.5, 4.2, and 10.2 ps, respectively) in the waveforms recorded at each location of the map. (b) Calculated 2D distribution of the normalized electric field (Ex) of the corresponding modes. (Reproduced from Mitrofanov and Harrington (2010), with the permission of OSA Publishing)

26 Optical Fibers in Terahertz Domain

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hollow-core and by extracting a 2D distribution at different time position in the 2D waveform (Mitrofanov and Harrington 2010). Indeed, the different mode velocities yield separated pulse packets in the waveform, at the fiber output. The first one corresponds to the mode HE11 , followed by the TE01 , HE12 , and HE13 modes. Besides, since the THz waves are guided in the hollow-core with a very low penetration in the polymer layer, propagated pulses experience a very low time broadening. The dispersion of the HE11 mode is estimated of about 0.08 ps/(mm) at 2.3 THz (i.e., “2 6 ps/(THzm)), for a PS-/Ag-coated hollow-core fiber (Dcore D 1.8 mm, t D 14 m) (Mitrofanov and Harrington 2010). Fibers with mechanical flexibility could be realized by using a polymer tube instead of a glass tube. Doradla et al. (2012) have coated the inner surface of a polycarbonate tube with PS/Ag layers for measuring the bending losses of such fibers. They have reported bend losses of about 0.3 dB/m (at 1.4 THz) for fibers with a hollow-core diameter from 2 to 4.1 mm that are bent around an aluminum disc of 6.4 cm diameter. However, the fibers are bent along an angle of 20ı only. This limitation is due to the degradation of the PS layer uniformity under the mechanical constraints created by the bend. In order to improve the flexibility capability of such fibers, Navarro-Cía et al. (2013) have fabricated an PS/Ag hollow-core fiber. An Ag layer with a thickness of 0.6 m and a PS layer of 10 m thick have been deposited on the inner surface of a silica capillary. The diameter of the hollow-core is only 1 mm for enabling a rather good mechanical flexibility of the fiber (up to a bending radius of 1.25 m). The first transmission window of the HE11 mode is from 2 to 5.5 THz. As shown in Fig. 17, THz radiations are propagated in the HE11 mode that is still the dominant mode when the fiber is bent, at least until a minimal radius of 1.25 m for which

Fig. 17 (a) Measured 2D distribution of the intensity in far field at the fiber output of THz radiation propagated through 1-m-long length of the Ag/PS hollow-core fiber (D D 1 mm and t D 10 m). (b) Bend losses measured at 2.85 THz for 1-m-long length fiber under different bending radius. (Reproduced from Navarro-Cía et al. (2013), with the permission of OSA Publishing)

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G. Humbert

the bending losses reach 1.0 dB/m at 2.85 THz. It is worth notifying that bend loss measurements have been realized with the polarization of the incoming THz radiations perpendicular to the plane of the bend, which is more favorable than the polarization parallel to the bending plane. However, reducing the core diameter for gaining in flexibility yields a higher attenuation coefficient of the HE11 , which one is experimentally measured below 10 dB/m from 2 to 2.85 THz and estimated below 3 dB/m from 3 to 5 THz. Even if specific applications requiring a short fiber length and a relative mechanical flexibility, this loss level is too large for most of the applications. Furthermore, this fiber structure that give some mechanical flexibilities is not transposable to lower frequencies. The flexibility is obtained with a core diameter of 1 mm, and bend measurements have been realized at 2.85 THz. For example, in the case of an operating frequency of 0.6 THz, the same guiding properties are obtained for a core diameter of 4.7 mm (by applying the same ratio D/œ D 9.5) leading to much less mechanical flexibilities. Despite its rather large attenuation coefficient, the mechanical flexibility of this fiber has been successfully exploited for delivering THz waves generated by a quantum cascade laser (QCL) over a distance of about 50 cm, within the HE11 mode (Vitiello et al. 2011). The coupling from the QCL to the fiber was realized through a copper-waveguide coupler that is robust to misalignment error, with a coupling efficient higher than 90% between the coupler and the fiber. This result demonstrates the advantages of such hollow-core THz fibers that exhibit broad transmission windows, a low GVD, waves guiding in the HE11 mode (Gaussianlike mode), and a relative mechanical flexibility (at high frequencies). Nevertheless, these fibers suffer from their large multimode guiding regime.

Hollow-Core Bragg Fibers Hollow-core Bragg fibers are composed of a hollow-core surrounded by a periodic stack of alternating high- and low-refractive index layers, as illustrated in Fig. 18. These fibers have been developed in the optical domain (Fink et al. 1999; Ibanescu

Refractive index

Λ

nh

dh

Hollow-core (D)

Dielectric layer, low refractive index (nl, dl)

Dielectric layer, high refractive index (nh, dh)

dl

nl nair 0

D/2

Radius

Fig. 18 Schematic illustration of the cross section and refractive index profile of a hollow-core Bragg fiber

26 Optical Fibers in Terahertz Domain

1047

et al. 2000) and then translated to THz frequencies. Light is confined and propagated in the hollow-core by multiple reflections in the periodic multilayer cladding that lead to constructive interferences of high reflectivity (Bragg reflections) in specific frequency bands (i.e., transmission windows). These bands exhibit photonic bandgaps properties (i.e., light propagation through the multilayer cladding is forbidden for any polarization) that reflect and confine the light in the hollow-core as a mirror (Fink et al. 1998, 1999; Temelkuran et al. 2002). The properties of this Bragg reflector depend on the thickness and the refractive index of the two layers and the number of stacked bilayers (number of periods). When the diameter of the hollow-core is significantly larger than the wavelength of the guided wave, the wave propagates almost parallel to the core-cladding interface (the effective index of the fundamental mode is almost equal to unity, neff 1) by consecutive reflections with grazing angles of incidence on the Bragg reflector cladding. In the idealistic case of an infinite number of bilayers, any polarization of light is completely reflected at the Bragg wavelength (œB ) that is given by this relation (Skorobogatiy and Dupuis 2007): œB D 2 .dh nQ h C dl nQ l /

(9)

p p p p with nQ h D nh C neff  nh C 1, nQ l D nl C neff  nl C 1, and nh , dh , nl , dl , the refractive index and the thickness of the high- and low-refractive index layers, respectively. The bandwidth of the bandgap is proportional to the refractive index contrast (nh  nl ), and it is maximized for the so-called quarter-wave reflector condition: dh nQ h D dl nQ l D

œB 4

(10)

In the realistic case of a finite size cladding, the light is not completely reflected by the Bragg reflector. The efficiency of the Bragg reflector increases with the refractive index contrast. Larger index contrast leads to larger reflections resulting to stronger light confinement in the hollow-core and thus to lower propagation losses. The size of the hollow-core has an effect on the propagation losses that decrease with a larger core diameter (D) according to the trend of œ2 /D3 . Temelkuran et al. have fabricated a hollow-core Bragg fiber with remarkable properties (Temelkuran et al. 2002). The Bragg reflector is composed of alternating layers of arsenic triselenide (As2 Se3 ) and poly(ether sulfone) (PES). They have reported an attenuation coefficient of about 0.95 dB/m at the wavelength 10.6 m, for a hollow-core diameter of 700 m. The large core diameter (D/œ D 70) reduces significantly the propagation losses, but it allows the propagation of an extremely large number of modes. It is worth to notify that this attenuation coefficient is much lower than the absorption coefficient of the materials of the Bragg reflector, which are 10 dB/m and 105 dB/m, respectively, for As2 Se3 and PES materials. This result demonstrates the efficiency of the Bragg reflector that enables light guiding in a hollow-core with materials that are 105 more absorbing than the attenuation

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coefficient. This property of the hollow-core Bragg fibers makes them very suitable for propagating waves in the THz domain, where most of the materials are very absorbing. The development of hollow-core Bragg fibers at THz frequencies is limited by the difficulties to find high- and low-refractive index materials with low absorption coefficients. Dupuis et al. have proposed to realize polymer/air bilayers or doped/undoped polymer bilayers (Dupuis et al. 2011). The polymer/air bilayers were obtained by randomly dispersing polymer powder on the top of a polymer film. The fiber was then fabricated by rolling the polymer film around a mandrel that is then removed to form the hollow-core. The powder particles act as spacers to form the air layers between the polymer layers. PTFE tape was then wrapped around the fiber for holding it. The fabricated polymer/air Bragg fiber has a hollow-core diameter of 6.73 mm surrounded by five bilayers consisting of a 254 m thick PTFE polymer and an air layer of about 150 m thick. As shown in Fig. 19a, the transmission spectrum of THz radiations propagated through 21.4 cm of the polymer/air hollow-core Bragg fiber is composed of clear transmission windows corresponding to the Bragg bandgaps. The measurements of the attenuation coefficient of this fiber are rather difficult. The large core enables the propagation of a large number of modes leading to transmission spectrum curves that depend on the modal interference between the excited modes. Slight fiber

Fig. 19 Measured transmission properties of a polymer/air hollow-core Bragg fiber (left column) and of a doped/undoped polymer hollow-core Bragg fiber (right column). (a, c) Normalized transmission (amplitude) spectrum of the fiber, with a cross-section photograph shown in the inset. (b, d) Attenuation spectrum of the fiber. (Reproduced from Dupuis et al. (2011), with the permission of OSA Publishing)

26 Optical Fibers in Terahertz Domain

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misalignments lead to significant variations in the transmission spectrum making the attenuation coefficient measurements poorly accurate. In this context, Dupuis et al. have measured the overall attenuation spectrum (including coupling losses) with a minimum attenuation lower than 0.028 cm1 (12 dB/m) at 0.82 THz (Cf. Fig. 19b). This value is much larger than the calculated attenuation coefficient of 3104 cm1 (0.13 dB/m) for the HE11 mode (Dupuis et al. 2011). This discrepancy emphasis the difficulty to measure accurately the attenuation spectrum of THz waveguides and the strong effect of intermodal couplings on the attenuation spectrum due to irregularities along the fiber and if the THz radiations are not carefully coupled to the HE11 only. For example, modes couplings to higher-order modes increase the attenuation coefficient by more than one order of magnitude, from 3104 cm1 (HE11 mode) to 7103 cm1 (3 dB/m, HE14 modes) (Dupuis et al. 2011). The doped/undoped polymer bilayers were realized by doping polyethylene (PE) polymer with TiO2 particles. The refractive index of PE was increased from 1.5 to 3 with a doping concentration of 80 wt.% of TiO2 . It is worth notifying that the absorption coefficient increases substantially and the thermomechanical properties of the polymer are also modified. The doped PE polymer has a higher viscosity and a much higher melting temperature than pure PE. A stack of doped/undoped PE films was hot-pressed together to consolidate the bilayer. The fiber is fabricated by rolling the stack around a mandrel that is then removed to form the hollow-core. The fabricated Bragg fiber has a hollow-core diameter of 6.63 mm surrounded by a six bilayers consisting of a 135 m thick high-index layer (of 80 wt.% TiO2 doped PE) and a 100 m thick low-index layers of undoped PE. The transmission spectrum of this fiber is less structured (Cf. Fig. 19c). Only one transmission band (in the low frequency side) is observable, for a fiber length of 22.5 cm. The overall attenuation coefficient spectrum is shown in Fig. 19d. The minimum attenuation coefficient is about 0.042 cm1 (18 dB/m) at 0.69 THz. As for the polymer/air Bragg fiber, this value is larger than the calculated attenuation coefficient of 103 cm1 (0.4 dB/m) for the TE01 mode. Furthermore, the large absorption coefficient of the doped polymer layer limits the formation of Bragg bandgap for HE modes that have wider field penetration into the Bragg cladding than the TE modes. In contrast to the polymer/air fiber, the lowest attenuated mode is the TE01 mode, which one requires specific conditions for properly coupling THz radiations into it. Furthermore, the fabrication of a TiO2 doped polymer film and the Bragg fiber is rather difficult to realize without irregularities or film defects (density inhomogeneity, cracks, etc.) that are additional factors of losses. Improvements of the fabrication processes and the use of less absorbing doping materials are two routes for improving the performances of this Bragg fiber that has the advantage to be more robust than the polymer/air fiber. In this prospect, Ung et al. have numerically study the compromise between the high refractive index contrast and material losses as function of TiO2 doping concentrations in polymer (Ung et al. 2011a). They have demonstrated two situations of interest: (i) low doping concentration (10 wt.% of TiO2 ) that leads to the minimum attenuation coefficient of HE11 mode with narrow transmission windows or (ii) very high doping

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G. Humbert

concentration (85 wt.% of TiO2 ) with large transmission windows and relatively low attenuation coefficient. These two doping concentrations are more favorable than intermediate values (between 30 and 70 wt.% of TiO2 ) where the attenuation coefficient and the transmission windows are both limited. Single-mode propagation in a Bragg fiber has been proposed by drastically reducing the size of the hollow-core and by carefully designing the Bragg reflector for guiding only the mode TE01 (Hong et al. 2017). The proposed Bragg fiber is composed of a hollow-core of 1.834 mm diameter surrounded by four bilayers TOPAS/air (with a thickness of about th D 77.5 m and tl D 1.154 mm, respectively) that are held together by thin TOPAS bridges (t D 15 m). The core diameter and the Bragg reflector are designed for matching the position of the second bandgap with the TE01 mode and for exhibiting bandpass properties (no confinement in the hollow-core) for the other modes, within the frequency range 0.8–1.2 THz. The confinement of the TE01 mode is realized in the second bandgap for enabling loss discriminations between the TE01 mode and the low order modes (HE11 , TM01 , HE21 ) by positioning a bandpass (band between the first and second bandgap) within the guiding region of these modes, as shown in Fig. 20a. As a result, the attenuation coefficient of the TE01 mode is lower than 1.2 dB/m from 0.85 to 1.15 THz (with a minimum of at 0.98 THz), which is more than one order of magnitude lower than the attenuation coefficient of the main competing mode HE11 (Cf. Fig. 20b). It is worth notifying this relatively low attenuation is obtained for a small hollow-core diameter that is only six times the wavelength (D/œ 6.1 at 1 THz). Nevertheless, the fabrication of this Bragg fiber is not demonstrated, probably due to the complexity due to complexity of the design (very thin bridges associated with thin TOPAS layers spaced by large air gaps).

Fig. 20 (a) Dispersion diagram of effective indices vs. frequency of modes confined within the hollow-core and the modes supported by the Bragg cladding (white area). Gray and black areas show domains where the Bragg cladding forbids wave extensions (i.e., bandgaps where no mode is confined in the cladding) for TE/HE and TM/EH modes, respectively. (b) Attenuation spectra of the first six modes guided within the hollow-core Bragg fiber. The fiber is composed of a hollowcore of 1.834 mm diameter surrounded by four bilayers TOPAS/air (with a thickness of about th D 77.5 m and tl D 1.154 mm, respectively) that are held together by thin TOPAS bridges (t D 15 m). (Reproduced from Hong et al. (2017), with the permission of IOP Publishing)

26 Optical Fibers in Terahertz Domain

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The recent progresses in the 3D printing technology might open new possibilities for fabricating THz fibers with complex design. Li et al. have fabricated a hollowcore Bragg fiber with a 3D printer (Li et al. 2017). The fiber is composed of a hollow-core surrounded by ten bilayers of high- and low-refractive index, namely, the printing resin (nh 1.654) and air that are held with additional micro-bridges. The thickness of each bilayer is about 512 m, with a predicted fundamental bandgap centered at 0.18 THz. The diameter of the hollow-core is reduced to 4.5 mm (D/œ D 2.6) for enabling the propagation of the HE11 mode only (at the expense of a higher attenuation coefficient) and achieving an effectively singlemode operation at the fundamental bandgap. The measured attenuation coefficient is about 0.12 cm1 (52 dB/m) within the narrow bandgap (45 GHz bandwidth centered at 0.18 THz), which is significantly smaller than the corresponding bulk absorption loss of the printing resin (’ 1 cm1 ). It is worth notifying that this process is limited to the fabrication of 25-mm-long length fiber. Nevertheless, this length is long enough for developing fiber-based sensors. Li et al. have realized a sensor by increasing the thickness of the first layer (high index) of the Bragg reflector in order to allow the confinement of a mode in this defect, by TIR mechanism at the interface hollow-core/high-index layer, and by the Bragg reflector (bandgap) (Li et al. 2017). This defect mode that exists within the bandgap frequencies is coupled with the HE11 mode at a specific frequency, which results as a sharp peak of losses in the transmission spectrum. The frequency of this peak is then sensitive to change in the hollow-core, especially around the defect layer. This property has been exploited for detecting added PMMA films on the inner core surface with a sensitivity of 0.1 GHz/m and for sensing ’-lactose monohydrate powder (deposited on the inner core surface) with a reliable detection of 3 m change in the analyte layer thickness. This performance which is among the best ever reported in THz sensing demonstrates the large potential of this strategy for developing fiber-based sensors.

Hollow-Core Pipe Fibers Hollow-core pipe fiber is simply a pipe composed of an air channel surrounded by a dielectric layer, as illustrated in Fig. 21. THz waves are confined and propagated in the air channel by the antiresonant reflecting mechanism. These THz fibers are analog to hollow-core photonic crystal fibers with a Kagomé lattice (Benabid et al. 2002; Pearce et al. 2007) and especially the simplified version composed of an air channel surrounded by a thin silica ring suspended in air by six silica struts (Gérôme et al. 2010). Antiresonant reflecting mechanism has been first developed by Duguay et al. (1986) for guiding light in planar optical waveguides. It acts in wave structures composed of one or several dielectric layers with thicknesses in the same order of magnitude of the wavelength of the confined wave. For hollow-pipe fibers, the dielectric layer of the pipe acts as a Fabry-Perot resonator; at the resonant frequencies, the wave crosses the layer, while it is reflected otherwise (under the

1052

t Transmission windows

Transmission

Fig. 21 Schematic representations of a hollow-core pipe fiber, of the two-dimensional distribution of the Poynting vector of the fundamental mode (HE11 ) confined into the air channel and of its transmission spectrum

G. Humbert

fm

fm+1

Frequency

Resonance frequencies

antiresonant condition) leading to wave confinement and propagation into the air channel. Transmission windows are delimited by the resonant frequencies that fulfill the following relation of the Fabry-Perot resonance conditions for a dielectric layer: fm D

mc p 2 t n2  1

(11)

with m the resonance order; t and n the thickness and the refractive index (real part) of the dielectric layer, respectively; and c the speed of light in vacuum. As illustrated in Fig. 21, the waves are guided in the air channel of the pipe within frequency-delimited transmission windows. Lai et al. (2009, 2010) have demonstrated the propagation of THz waves in the air channel of Teflon or polymethacrylate of methyl pipes. These commercially available pipes are composed of a hollow-core diameter (D) of 9 mm and a dielectric layer thickness (t) of 1 or 0.5 mm. The measured free-space coupling coefficient from the source to the fiber is around 40% in average with a maximum of 84%. The attenuation coefficient measured by the cutback method with a pipe length up to 3 m is as low as 0.0008 cm1 (’ D 0.35 dB/m) below 1 THz. The attenuation coefficient is proportional to the hollow-core diameter following a 1/D4 dependency. It is worth notifying that this trend is different than the 1/D3 dependence of a hollowcore surrounded by an infinite thick dielectric layer or dielectric pipe with a thick layer (t  œ). Therefore, lower attenuation coefficient of the fundamental mode HE11 is obtainable with large core size. Nevertheless, high-order modes are also propagated into the air channel. The modes propagated in the air channel pipe by the antiresonant reflecting guiding mechanism are not pure modes. They are leaky modes. In contrast with the modes of classical optical fibers (based on TIR guiding mechanism), the modes exhibit propagation losses, and they do not have a cutoff frequency. The distributions

26 Optical Fibers in Terahertz Domain

a

3) TM01

5) HE21y

b 2) HE11y

4) HE21x

6) TE01

1000

Attenuation coefficient (dB/m)

1) HE 11x

1053

TM01 HE21

100

TE01

Δneff = 2.6 .10

-2

HE11

ni = 10

-3

Dcore = 9 mm

10

ni = 0 ni = 1e-3

TM01 HE21 Δneff = 3.2 .10

Dcore = 3 mm

1

-3

ni = 0 ni = 1e-3

0.96

TE01

0.97

0.98

0.99

HE11

1.00

Real(Effective index) Fig. 22 (a) 2D distribution of the Poynting vector of the fundamental mode (HE11 ) and first higher-order modes. (b) Attenuation coefficient and effective index (real part) of the fundamental mode (HE11 ) and first higher-order modes, for dielectric pipes (n D 1.4) of different core diameters, with or without material absorption

(in two dimensions) of the Poynting vector of the first six modes (with the highest effective index) confined in the air channel are shown in Fig. 22a. Their attenuation coefficients are plotted in Fig. 22b versus their effective indices for a core diameter of D D 9 or 3 mm. The leaky nature of the modes is highlighted by their large attenuation coefficients in the idealistic case of a dielectric layer without material absorption. As already mentioned, the size of the core has a strong effect on the attenuation coefficient of the modes. For example, the attenuation coefficient of the fundamental mode (HE11 ) is reduced by almost two orders of magnitudes (from 92 to 1.5 dB/m) when the core diameter is increased from D/œ D 4 to 12. This trend is similar for higher-order modes. Therefore, increasing the core size is a good strategy for reducing the propagation losses, but without improving the discrimination between the fundamental and high-order modes. For the TE01 mode, the mode with the closest attenuation coefficient to HE11 mode, the ratio ’TE01 /’HE11 decreases from 2 to 1.2, for a core diameter of 3 and 9 mm, respectively. Furthermore, larger core size increases the effective index of all modes toward unity (nair D 1) pushing the higher-order modes closer to the fundamental modes. For example, the difference between the effective index of the HE11 and TE01 modes (neff D neffHE11  neffTE01 ) decreases from 2.6102 to 3.2103 for a core diameter of 3 and 9 mm, respectively. In consequence, couplings between the fundamental mode and higher order ones are more likely to happen with larger core size leading to multimode propagations and additional losses from intermodal couplings. The effect of material absorption from the dielectric layer is rather small due to the confinement of the waves in the air channel with low penetration in the

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dielectric layer. As shown in Fig. 22, the material absorption is reduced by 45 times for a core diameter of 9 mm. The ratio ’mat /’HE11 is mainly governed by the attenuation coefficient of the pipe without material absorption. Indeed, adding a material absorption of ’mat D 72.76 dB/m at 400 GHz (corresponding to an imaginary part of refractive index ni D 103 ) increases the attenuation coefficient of the mode HE11 by only 1.04 or 1.10 for a core diameter of 3 and 9 mm, respectively. For a material absorption of ni D 102 , this ratio increases only to 1.19 and 1.35 for a core diameter of 3 and 9 mm, respectively. This leads to a reduction of the material absorption by 260 times for a core diameter of 9 mm. It is worth notifying that the effect of material absorption is slightly lower for smaller core diameter. But this has to be balanced by the large augmentation of the attenuation coefficient of the modes (even with ’mat D 0), which ones could be higher than the material absorption coefficient, as it is the case for a diameter of 3 mm. For numerous optical fibers, bends generate additional transmission losses. In the case of hollow-core pipe fiber, bend losses of about 2.6 dB/m (0.006 cm1 , at 420 GHz) have been reported for a bend radius of 60 cm applied on a pipe fiber (D D 9 mm and t D 0.5 mm) (Lu et al. 2010). In these fibers, the bend losses are function of the spectrum of the transmission bands. They are larger at frequencies close to the resonant frequencies and smaller in the middle of the transmission bands where the wave confinements are stronger (Lu et al. 2010). Furthermore, the bend losses increase with the thickness of the dielectric layer (t) and decrease with larger core size. Nevertheless, Lai et al. (2014) have demonstrated that for a given bend radius, there is a critical core diameter above which bend loss reductions are less efficient. Chih-Hsien Lai et al. explain this behavior with the picture of ray propagations. At a given frequency and bend radius, the waves are bounced off the wall of the air channel at both inner and outer core-wall interfaces (in the plan of the bend). When the core diameter increases, the distance between two consecutive reflections also increases. As a result, the number of reflection inside the waveguide reduces, and thus the loss resulting from the reflection decreases. Above the critical core diameter, the waves are reflected only at the outer core-wall interface. This regime is similar to whispering gallery modes that are confined by multiple reflections at the outer boundary of a sphere or a toroid. This regime is more likely to appear with small bend radius, large air core diameter, or higher operating frequency. Since the transmission windows are delimited by the resonant frequencies, the refractive index and the thickness of the dielectric layer define their bandwidth (œ) with the following relation: f D

c p 2 t n2  1

(12)

A lower refractive index and a thinner wall lead therefore to broader transmission windows. Broad transmission windows of 0.36 and 0.58 THz have been demonstrated with silica pipes composed of a layer thickness of 250 and 155 m, respectively, and an outer diameter of 5 mm (Nguema et al. 2011). Transmission

26 Optical Fibers in Terahertz Domain

1055

Fig. 23 (a) Fourier amplitude spectra of measured THz pulses after propagation in free space (reference) or through a 40-cm-long length silica pipe (t D 155 m, D D 5 mm). (b) Measured THz pulses after propagation through a 50-cm-long length “commercial drinking straw”: (red curve) Fourier amplitude spectrum, (blue curve) measured effective index of the propagated THz fields

spectrum of the silica pipe with a layer thickness of 155 m is shown in Fig. 23a. Much wider transmission windows could be obtained with lower refractive index materials than silica (n 1.95) such as polyethylene (PE, n 1.5). As shown in Fig. 23b, a transmission window of 1.02 THz is obtained with a simple “commercial drinking straw” composed of a 120 m thick PE layer and an outer diameter of 6 mm. THz wave guidance through the air channel of the straw is confirmed by the evolution of the measured effective index of the electromagnetic field (Cf. Fig. 23b). The values are below unity, and the curve presents the characteristic shape within the transmission window and with variations closed to the resonance frequencies (at least for the first one).

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Fig. 24 Simulated attenuation coefficient and group velocity dispersion of the HE11 mode propagated into the air channel of a thin PE pipe (t D 143 m, n D 1.45, D D 6 mm)

It is worth notifying that the effective index is tangent to unity for larger frequency leading to lower GVD values. As shown in Fig. 24, the GVD of the HE11 mode propagated into the air channel of a thin PE pipe (t D 143 m, n D 1.45, D D 6 mm) is strongly affected by the degradations of the guiding conditions close to the resonant frequencies leading to sharp variations of the GVD from negative to positive values. The GVD is smaller in the second transmission windows. At larger frequencies, the size of the core compared to the wavelength size is larger. As a result, the HE11 mode could propagate more parallel (to the propagation axis) with a larger distance between two consecutive reflections at the dielectric layer interface, leading to properties closer to the ones of a wave propagating in air. The ratio D/œ is two times larger in the second windows than in the first one. As a consequence, the GVD is below ˙2 ps/(THzm) in a broader frequency range, in the second window (œ D 700 GHz, from 1.2 to 1.9 THz) than in the first one (œ D 160 GHz, from 0.7 to 0.86 THz). In comparison to other THz fibers, their broad transmission windows, relatively low attenuation coefficient, and very low GVD make these fibers very attractive especially for pulse propagations. However, these performances are obtained for a large core diameter (D > 20œ) that induces favorable conditions for multimode propagations and additional losses from intermodal couplings (as shown in Fig. 22), leading to drastic conditions for coupling THz radiations to only the HE11 mode of the air channel. Even if most of the work reported on hollow-core THz pipe fibers used commercial polymer pipes, some THz pipes have been especially developed by drawing down thin-wall silica capillaries (from a silica tube) with optical fiber

26 Optical Fibers in Terahertz Domain

1057

fabrication facilities (Nguema et al. 2011), by fabricating thin-wall chalcogenideglass capillaries with a double crucible glass drawing technique (Mazhorova et al. 2012), or also by drawing down thin-wall PMMA capillaries (from a PMMA tube) with low-temperature fiber drawing tower (Xiao et al. 2013a). Xiao et al. have also fabricated a self-supporting PMMA pipe by drawing down a PMMA tube filled with seven PMMA capillaries (Xiao et al. 2013b). The fiber is composed of a thin hexagonal inner pipe delimiting a hollow-core that is supported by six thin walls attached on the outer pipe. By this process, they have been able to reduce the thickness of the inner pipe down to 20 m leading to broad transmission windows in the highest THz frequency range (œ 6.5 THz). Finally, square or rectangular polymer pipes have been fabricated by sticking four PE or PMMA strips (Lu et al. 2011a). Rectangular pipes have been developed for studying and demonstrating some polarization sensitivity of the attenuation coefficient that could be controlled by adjusting the structure of the rectangle. Furthermore, directional coupler composed of two squares or rectangular pipes have been demonstrated by simply placing both pipes in contact side by side (Lu et al. 2011b). A hollow-core THz pipe fiber has been successfully integrated into a reflective THz fiber-scan imaging system for in situ and dynamically monitoring the chemical reaction of hydrochloric acid (HCl) and ammonia (NH3 ) vapors to generate ammonium chloride (NH4 Cl) aerosols in a sealed chamber (You and Lu 2016). A simple Teflon pipe has been used as a sensing and image-scanning arm. The bending capability of the reflective THz pipe fiber-based scan system has been demonstrated by imaging and identifying an array of tablets. Applications of hollow-core THz pipe fibers have been also demonstrated for high-performance sensing by tracking the shift of the resonant frequencies, when a sub-wavelengththick molecular overlayer (with different Carbopol aqueous solutions) is adhered to the inner core surface of a tube (You et al. 2010a), when various powders are loaded on the outer surface of the pipe (composed of an absorptive layer), or when different vapors are inserted into the air channel of the pipe (You et al. 2012). The guiding mechanism and properties of hollow-core THz pipe fibers are the basement of others hollow-core THz fibers, even if they have been developed independently. These THz fibers are presented in the following sections.

Kagome Hollow-Core Photonic Crystal Fibers Kagome hollow-core PCF have been intensively developed in the optical domain for exploiting their broadband transmissions. These fibers are analog to hollow-core pipe fibers; they are composed of concentric thin hexagonal inner pipes supported by thin walls forming a Kagome lattice. Light is confined and propagated in the hollowcore by antiresonance guiding mechanism with inhibited coupling to cladding modes, if the wall connections are free of apex (where parasitic couplings could appear due to cladding mode confinement) (Couny et al. 2007). The properties of Kagome hollow-core PCF are therefore similar to the ones of the hollow-core pipe fibers. As shown in Fig. 25a, the attenuation spectrum

1058

G. Humbert

Fig. 25 (a) Numerically calculated attenuation spectra of the fundamental mode of a Kagome hollow-core PCF composed of two or four layers, together with that of a hollow-core pipe fibers composed of one or two layers. Schematics of the fibers structures are shown in the insets. Black and gray areas represent glass and white areas represent air. (Reproduced from Pearce et al. (2007), with the permission of OSA Publishing). (b) Optical micrograph of the cross section of a PMMA Kagome fiber showing the hollow-core and the innermost four of the six rings of the cladding. (Reproduced from Anthony et al. (2011b), with the permission of OSA Publishing)

is smaller than the one of a hollow-core pipe, due to the addition of at least one antiresonant layer (i.e., a thin hexagonal layer) (Pearce et al. 2007). Nevertheless, the supporting thin walls add some parasitic couplings between the core mode and cladding modes that perturb the attenuation spectrum with high attenuation peaks.

26 Optical Fibers in Terahertz Domain

1059

Anthony et al. have fabricated different Kagome hollow-core PCF in PMMA by drawing a stack of PMMA tubes (filled within a larger PMMA tube) down to fibers with core and outer diameters around 2 and 6 mm, respectively (Cf. Fig. 25b) (Anthony et al. 2011b). The hollow-core was formed by removing seven tubes from the stack composed of six hexagonal rings of tubes (in a triangular lattice) to form the cladding. Even if the cladding is not a perfectly Kagome lattice, they have demonstrated the propagation of THz waves with an attenuation coefficient about 0.6 cm1 (260 dB/m) in the frequency range from 0.65 to 1.0 THz, corresponding to a reduction of about 20 times of the absorption coefficient of PMMA material. Yang et al. have fabricated a Kagome hollow-core PCF in polymer material by 3D printing (Yang et al. 2016). The fiber is composed of a hollow-core of 9 mm diameter surrounded by two layers of air holes with a diameter of 3 mm and a wall thickness of 0.35 mm. The maximum fiber length was 30 cm. The cladding design is more a triangular lattice of connected capillaries than a Kagome lattice. Nevertheless, they have demonstrated the propagation of THz waves with an average attenuation coefficient of 0.02 cm1 (8.7 dB/m) within the frequency range from 0.2 to 1.0 THz with the minimum attenuation about 0.002 cm1 (0.87 dB/m) at 0.75 THz. Even if the Kagome-lattice cladding protects the waves guided in the core to external perturbations, the large outer diameter of such fiber could be reduced since most of the confinement is realized by the two first layers as shown on Fig. 25a. In this prospect, Wu et al. have proposed a numerical study on the reduction of the overall fiber diameter by eliminating the outermost cladding layers for the resulting fibers to be practical and flexible (Wu et al. 2011).

Tube Lattice Hollow-Core Fibers Tube lattice hollow-core THz fibers are a new family of hollow-core fibers based on antiresonance guiding mechanism with inhibited coupling to cladding modes (Vincetti 2010). These fibers are composed of a hollow-core surrounded by a cladding formed by a periodic arrangement of tubes in a triangular lattice (Cf. Fig. 26a, b). The properties of these fibers are similar to the ones of the hollowcore pipe fiber or Kagome hollow-core PCF that are characterized by broadband transmission windows. The wave confinement and propagation in the hollow-core is obtained by inhibiting the couplings between the core modes and the cladding modes. The field overlap between the core and the cladding modes is reduced by the tube lattice cladding that is free of node, avoiding localization of cladding modes. As in pipe or Kagome fibers, the transmission windows are limited by sharp propagation losses at the cutoff frequencies of cladding modes where core modes are coupled to leaky cladding modes. These frequencies are well approximated by those of a single tube from the cladding that is defined by its refractive index, wall thickness, and external diameter (Vincetti and Setti 2010). Wider

1060

G. Humbert

D

a

b

t

c

Fig. 26 Schematics of the tube lattice hollow-core THz fiber composed of (a) two layers of tube lattice, (b) one layer of tube lattice, and (c) one layer of tube lattice with an octagonal symmetry

transmission windows are obtained for a tube with a thinner wall and a large diameter (compared to the wavelength). In this condition, the cutoff frequencies are similar to those of an infinite slab that are following the resonance conditions of a Fabry-Perot (Eq. 11). As for a pipe or a Kagome fiber, the transmission windows depend therefore on the wall thickness of the tubes that form the cladding. This property enables the fabrication of THz fibers with broader transmission windows than that of pipe fiber, since it is easier to fabricate several very thinwall tubes with a relative small diameter (1 mm) than one with a large diameter (several millimeters). Furthermore, a large thin-wall pipe is less flexible than a bundle of small diameter thin-wall tubes, making the tube lattice hollow-core fiber an interesting design for developing low-loss flexible THz fibers. The tube lattice cladding presents another interest for reducing the propagation losses. In this configuration, a part of the external boundary of each tube in the first ring of the cladding delimits the hollow-core that could be represented as a large circle composed of negative curvatures (half perimeter) from each tube. These negative curvatures minimize the spatial overlap between the core modes and the cladding modes (in the tube lattice) leading to a better confinement and lower attenuation coefficient of the core modes (Wang et al. 2011; Pryamikov et al. 2011). An attenuation coefficient of the HE11 mode as low as 0.07 dB/m at the frequency of 1.9 THz (and lower than 0.1 dB/m over a range of about 800 GHz) has been numerically predicted for a fiber composed of 12 Teflon tubes (of 1 mm diameter and wall thickness of 44 m) surrounding an hollow-core (Vincetti 2010). The core shape corresponds to seven missing tubes packed in a triangular lattice, leading to a core diameter smaller than 3 mm. The use of the triangular lattice in the fiber design comes from the fabrication process of PCF that is based on stacking capillaries or tubes in a triangular lattice. The tube lattice fibers composed of only one ring are not restricted to this design. A tube lattice fiber with octagonal symmetry (Cf. Fig. 26c), i.e., 8 tubes surrounding

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the hollow-core instead of 12, could improve the loss discrimination between the HE11 mode and TE01 mode, by a factor of about 2.6 (Vincetti et al. 2010). This is attributed to better phase-matching conditions between the high-order core modes and cladding modes. Following this development, Setti et al. (2013) have fabricated a hollow-core tube lattice THz fiber composed of height PMMA tubes with an external diameter of about 1.99 mm and a wall thickness around 252 m, leading to a core size around 3.42 mm (Cf. Fig. 27a). Small pieces (L 5 mm) of jacket PMMA tubes have been added each 10 cm on the fiber structure for ensuring its mechanical stability. They have measured the propagation of THz waves within two transmission windows (0.3–0.5 THz and 0.6–0.95 THz) with an attenuation coefficient of 0.3 dB/cm at 0.375 THz and 0.16 dB/cm at 0.828 THz, corresponding to a reduction to the bulk PMMA material by 31 and 272 times, respectively. These measurements and the simulated attenuation spectra of the fundamental core mode for different material absorption coefficient of PMMA (imaginary part of the refractive index) are shown in Fig. 27c.

Fig. 27 (a) Picture of the tube lattice hollow-core THz fiber fabricated by Setti et al. (b) Simulated Poynting vector distribution of the fundamental core mode (A) and different cladding modes confined in tube holes (B) or in tube walls (C, D). (c) Measured attenuation coefficient of the THz waves propagated through the THz fiber (green dots) and numerical results of attenuation coefficient of the fundamental core mode with different values of the imaginary part of the refractive index of the tube material. (d) Simulated Poynting vector distribution of the fundamental core mode for different fiber bending radius (A, Rb 21 cm; B, Rb 15 cm; C, Rb 7 cm; D, Rb 12 cm) at frequency where couplings to cladding modes appear, leading to extra losses. (Reproduced from Setti et al. (2013), with the permission of OSA Publishing)

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As previously mentioned, this fiber structure is favorable for broadband transmissions and bending. Setti et al. have experimentally demonstrated a clear reduction in the transmission spectrum of the first or second window for a bend radius of 10 or 30 cm, respectively, showing that the lower frequency transmission window is the most robust against bends (Setti et al. 2013). Bends induce a reduction of the transmission windows by a shift of the higher frequency edges to lower frequency. This shift is induced by a variation of the cutoff frequencies of the cladding modes. The bend could be considered following the conformal mapping technique, where the bent fiber is analyzed as a straight fiber with negative and positive variations (in the bend direction) of the refractive index, respectively, in the inner and outer bend curvature, and with amplitude depending on the bend radius. In this picture, smaller bend radius leads to higher refractive index in the outer bent curvature and therefore to smaller cutoff frequencies of the cladding modes that delimit the transmission windows. Furthermore, when the bend radius is small, extra losses due to the resonances between the core mode and hole modes appear (Cf. Fig. 27d). The refractive indices inside the cladding holes change according to their relative position along the bend direction and the bend radius. In particular, the effective indices of the hole modes of the tubes increase approaching the one of the core mode until the phase matching condition is reached, causing the extra losses (Setti et al. 2013). The mechanical flexibility of this kind of fiber has been successfully exploited as a probe in an endoscope geometry and used to guide radiation to and from a sample for enabling THz hyperspectral imaging by scanning the end face of the probe in front of the sample (Lu and Argyros 2014). The reduction of the attenuation coefficient was realized by replacing the PMMA tubes with Zeonex polymer tubes (Lu et al. 2016). The absorption coefficient of Zeonex cycloolefin polymer is about of ’ D 0.14–0.3 cm1 for frequencies between 0.2 and 1.2 THz. Lu et al. have measured an attenuation coefficient of about 5 dB/m around 0.275 THz. They have also estimated from simulation an attenuation coefficient below 0.1 dB/m in the frequency range 0.8–1.2 THz for a fiber composed of a hollow-core of about 3.3 mm diameter surrounded by ten Zeonex tubes with a wall thickness of 92 m (Lu et al. 2016). These results are among the lowest losses reported in this frequency range, making the tube lattice hollow-core THz fibers one of the best THz fibers. Further reduction of the attenuation coefficient might be achievable by exploiting advanced design of tube lattice hollow-core fibers developed in the optical domains. Belardi and Knight (2014) have demonstrated from a simulation study the reduction of the attenuation coefficient by several orders (more than two orders) of magnitudes by adding one or two tubes within each tube of the cladding (composed of only a single layer), with the same thickness and attached to the cladding jacket at the same azimuthal position. The attenuation coefficient could be further reduced by at least one order of magnitude by adding some air gap between the main tubes (with or without added smaller tubes) of the cladding (Poletti 2014). Nevertheless, the interest of these improved designs has to be evaluated in THz domain by considering the large absorption coefficient of the tube material.

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Conclusion The degradation of the conductivity of metals and the low transparency of most dielectric materials in the THz domain call for innovative fiber designs that are mostly inspired from designs of specialty optical fibers. Porous fibers and tube lattice hollow-core fibers are part of fiber designs that originates from THz domain. They have been developed by taking advantages of the larger size of fiber design features at THz frequencies (than at optical ones). It is worth to notify that the tube lattice hollow-core fibers have been then successfully developed in the optical domain. This achievement emphasizes another interest of THz fibers for developing and testing new fiber designs that could be benefit for specialty optical fibers in the optical and infrared domains. Numerous fiber desings that have been investigated open been investigated that open various strategy for improving the performances of THz fibers. Even if the attenuation coefficient of THz fibers does not enable to deliver THz radiations over long fiber lengths, the performances of THz fibers become satisfactory for developing components and systems. These emerging realizations exploit the properties of specific THz fibers depending on the targeted applications. Solid-core fibers have the advantages of (i) single-mode propagation (HE11 mode, with a Gaussian intensity distribution), (ii) a large fraction of evanescent power, and (iii) a GVD curve that depends on the fiber design. On the other side, they are sensitive to external perturbations, have a limited transmission windows (expected for the PCF), their properties and designs strongly depend on the refractive index and absorption coefficient of the fiber material (especially for the PCF), and, the mean GVD value is rather high (limiting pulse propagations to few tens of centimeters). The confinement of the THz waves in a hollow-core yields a low GVD, a rather low attenuation coefficient, and a low sensitivity to external perturbations (except for hollow-core pipe fiber). Nevertheless, hollow-core THz fibers have limited transmission windows, and they could guide a large number of higher-order modes (in addition to the fundamental one) if THz radiations are not carefully coupled to the fundamental mode. It is difficult to fairly compare the performances of each hollow-core THz fiber, since the reported properties depend on the core size and the operating wavelength (i.e., D/œ), on the materials used, on the fabrication processes employed, and on the measurement methods. Even if some fiber designs require complex fabrication process that could affect the performances of the fibers, a comparison of the hollow-core THz fibers should be realized by simulating the different fiber designs with the same core size, refractive index (complex), and operating wavelength. Besides, the hollow-core fibers could lead to a very low attenuation coefficient by increasing the core size, but at the expense of stronger intermodal couplings to higher-order modes. Future designs of hollow-core fibers might therefore decrease the intermodal couplings by increasing the losses and phase differences between the fundamental and high-order modes, in order to enable an effective single-mode propagation.

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In order to sustain the intense efforts on the development of THz devices, upcoming developments of THz fibers should investigate novel designs (to reduce the attenuation coefficient and to improve bending capabilities) and new materials (with reduced absorption coefficient). The developments of fabrication processes dedicated to THz fiber specifications (materials and size), and of rigorous characterization protocols of THz fibers (compatible with low attenuation coefficient measurements) are also required. These underlying challenges make the development of THz fibers a very stimulating research area with myriad of potential applications.

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Optical Fibers for Biomedical Applications

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Concepts of Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Light-Guiding Principles in Conventional Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ray Optics Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modal Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graded-Index Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance Characteristics of Generic Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . Optical Fibers Used in Biophotonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conventional Solid-Core Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specialty Solid-Core Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Double-Clad Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hard-Clad Silica Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coated Hollow-Core Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photonic Crystal Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plastic Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Side-Emitting or Glowing Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Middle-Infrared Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optical Fiber Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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This chapter describes the various types of optical fibers that are used for precise delivery of light to specific biological tissue areas and for collection of reflected, scattered, or fluorescing light resulting from the light-tissue interaction. First, the Introduction gives some background information on biophotonics as it applies to the human body. Next, section “Basic Concepts of Optical Fibers” discusses G. Keiser () Boston University, Boston, MA, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 G.-D. Peng (ed.), Handbook of Optical Fibers, https://doi.org/10.1007/978-981-10-7087-7_35

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the fundamental principles of how conventional fibers guide light along the fiber. Here the term “conventional” refers to the structure of optical fibers that are used widely in telecom networks. This discussion will be the basis for describing how light propagates in other optical fiber structures. In addition, section “Basic Concepts of Optical Fibers” describes the necessary performance characteristics of optical fibers for applications in specific spectral bands. Based on this background information, section “Optical Fibers Used in Biophotonics” then describes categories of optical fiber structures that are appropriate for use in different biophotonic applications.

Introduction The category of photonics that deals with the interaction between light and biological material, such as human tissue, is known as biophotonics or biomedical optics (Popp et al. 2011; Keiser 2016; Ho et al. 2016). The result of this light-tissue interaction includes reflection, absorption, fluorescence, and scattering manifestations of light by tissue samples. These interactions are widely used in facilities such as research laboratories, pathology departments, hospitals, and health clinics to analyze the characteristics and health conditions of biological tissues and to carry out therapy on diseased or injured tissue. Note that commonly the word human tissue, or simply tissue, is used to designate all categories of human biological material including components such as flesh, bones, blood and lymph vessels, and body fluids. From a general viewpoint, biophotonics involves the detection, reflection, emission, modification, absorption, creation, and manipulation of photons as they interact with biological tissue components. The biomedical areas of interest include (a) imaging techniques of biological elements ranging from cells to organs, (b) noninvasive measurements of biometric parameters such as blood oxygen and glucose levels, (c) light-based treatment of injured or diseased tissue, (d) detection of injured or diseased cells and tissue, (e) monitoring of wound healing progress, and (f) surgical procedures such as laser cutting, tissue ablation, and removal of cells and tissue. The use of various types of optical fibers is attractive for such biophotonic applications because fibers allow pinpoint illumination of tissue areas in order to investigate the structural, functional, mechanical, biological, and chemical properties of biological material and systems. In addition, optical fibers play a key role in biophotonic methodologies that are being used extensively to investigate and monitor the health and well-being of humans. Among the numerous diverse biophotonic applications of optical fibers are imaging, spectroscopy, endoscopy, tissue pathology, blood flow monitoring, light therapy, biosensing, biostimulation, laser surgery, dentistry, dermatology, and health status monitoring. As shown in Fig. 1, the spectral bands of interest for biophotonics range from the mid-ultraviolet (about 190 nm) to the mid-infrared (about10.6 m) regions. Numerous applications are in the visible 400–700 nm spectrum, because of the

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Absorption coefficient (cm–1)

Diagnostic or KrF ArF 249 nm therapeutic window 193 nm

CO2 10.6 µm

Melanin

1000 100

Er: YAG 2940 nm

Whole blood

Proteins

Water 10

Epidermis

1 Water 0.10

0.20

0.50

1.00

2.00

5.00

10.0

Wavelength (µm)

Fig. 1 Absorption coefficients of several major tissue components as a function of wavelength with example light source emission peaks (After Keiser 2016)

relatively low absorption of light in this band compared to other spectral regions. As Fig. 1 shows, quite different levels of photon absorption occur in the spectral regions that are of interest to biophotonics. Thus, a broad range of diverse photonic tools and both standard commercially available and custom-made optical fibers that operate efficiently in designated lightwave spectral bands are employed in biophotonics. For example, spectral regions with low absorption (such as the visible band) are ideal for imaging relatively deep into tissue. The ultraviolet and infrared spectral bands exhibit strong absorption and thus are suitable for cutting and removal of tissue material. The unique physical and light transmission properties of optical fibers enable them to help resolve challenging biomedical implementation issues. These challenges include: (a) Collecting emitted low-power light from a tissue specimen and transmitting it to a photon detection and analysis instrument (b) Delivering a wide span of optical power levels to a tissue region during different categories of therapeutic healthcare sessions (c) Accessing a diagnostic or treatment area within a living being with an optical detection probe or a radiant energy source in a manner that is the least invasive to the tissue host Consequently, diverse types of optical fibers are finding widespread use in biophotonics instrumentation for clinical and biomedical research applications. In terms of transmission characteristics, flexibility, strength, and size, each optical fiber

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structure has certain advantages and limitations for specific uses in different spectral bands (Keiser et al. 2014). Therefore, it is essential that biophotonics researchers and designers of clinical biomedical optical fiber-based tools know what fiber type is best suited for a certain application.

Basic Concepts of Optical Fibers This section discusses the fundamental principles for light guiding in two categories of conventional solid-core fibers. These discussions will be a guide for understanding light-guiding mechanisms in other optical fiber structures. Several categories of optical fiber structures consisting of various materials are appropriate for use at different wavelengths in biomedical research and clinical practice. These include conventional and specialty solid-core fibers, double-clad fibers, hard-clad silica fibers, internally coated hollow-core fibers, photonic crystal fibers, polymer fibers, side-emitting and side-firing fibers, middle-infrared fibers, and fiber bundles. The fiber materials that are employed at different wavelengths include standard silica, UV-resistant silica, halide glasses, and polymer materials. Table 1 summarizes the characteristics of these optical fibers. The biophotonic applications in this table have been designated by the following three general categories with some basic examples: 1. Light care: healthcare monitoring, laser surfacing or photorejuvenation 2. Light diagnosis: biosensing, endoscopy, imaging, microscopy, spectroscopy 3. Light therapy: ablation, photobiomodulation, dentistry, laser surgery, oncology

Light-Guiding Principles in Conventional Fibers A conventional solid-core fiber is a dielectric waveguide that operates at optical frequencies (Keiser 2015). This fiber waveguide is normally cylindrical in form. It confines and guides electromagnetic energy at optical wavelengths within its surfaces. The propagation of light along an optical waveguide can be described by means of a set of guided electromagnetic waves called the modes of the waveguide. Each guided mode is a pattern of electric and magnetic field distributions that is repeated along the fiber at periodic intervals. Only a certain discrete number of modes can propagate along the optical fiber. These modes are those electromagnetic waves that satisfy the homogeneous wave equation in the fiber and the boundary condition at the waveguide surfaces. Figure 2 shows a schematic of a conventional optical fiber. This waveguide structure is a cylindrical silica-based glass core surrounded by a glass cladding that has a slightly different composition. The core has a refractive index n1 and the cladding has a slightly lower refractive index n2 . Encapsulating these two layers is a polymer buffer coating that protects the fiber from adverse mechanical and environmental effects. The refractive index of pure silica varies with wavelength

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Table 1 Major categories of optical fibers and their applications to biomedical research and clinical practice (After Keiser 2016) Optical fiber types Conventional solid-core silica fibers Specialty solid-core fibers

Multimode

Single-mode Photosensitive

UV-resistant

Bend-loss insensitive Polarizationmaintaining Double-clad fibers Hard-clad silica fibers

Coated hollow-core fibers Photonic crystal fibers Plastic optical fibers or polymer optical fibers Side-emitting fibers and side-firing fibers Mid-infrared fibers

Optical fiber bundles

Characteristics Multimode propagation; carry more optical power

Biophotonic applications Light diagnosis; light therapy

Single-mode propagation High photosensitivity to UV radiation; FBG fabrication Low UV sensitivity and reduced attenuation below 300 nm High NA and low bend-loss sensitivity High birefringence and preserve the state of polarization Single-mode core and multimode inner cladding Silica glass core with thin plastic cladding; increased fiber strength; high-power transmission Low absorption for mid-IR and high optical damage threshold Low loss; transmit high optical power without nonlinear effects Low cost; fracture resistance; biocompatibility Emit light along the fiber or perpendicular to the fiber axis Efficient IR delivery; large refractive index and thermal expansion Consist of multiple individual fibers

Light diagnosis Light care; light therapy Light diagnosis

Light therapy Light diagnosis

Light diagnosis Light diagnosis; light therapy

Light therapy

Light diagnosis; light therapy Light diagnosis

Light therapy

Light diagnosis; light therapy Light diagnosis

ranging from 1.453 at 850 nm to 1.445 at 1550 nm. By adding certain impurities such as germanium oxide to the silica during the fiber manufacturing process, the index can be changed slightly. Design variations in the optical fiber material and the diameter of the conventional solid-core fiber structure dictate how a light signal is transmitted along a fiber. These variations also influence how the lightwaves in the fiber respond to

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Core diameter 2a

Elastic protective buffer coating

Cladding n2 < n1

Core n1

Fig. 2 Schematic of a conventional silica fiber structure

Nominal dimensions

Index profiles n2

n1

125 mm (cladding)

2a

8–12 mm (core)

n2

n1

125–400 mm (cladding) 50–200 mm (core)

2a

n2

Step-index single-mode fiber

n1 r=a r

r=0

Step-index multimode fiber

125–140 mm (cladding) 50–100 mm (core)

Graded-index multimode fiber

Fig. 3 Comparison of conventional single-mode and multimode step-index and graded-index optical fibers (After Keiser 2016)

environmental perturbations, such as stress, bending, and temperature fluctuations. Changing the material composition of the core gives rise to two commonly used fiber types, as shown in Fig. 3. In the first case, the refractive index of the core is uniform throughout and undergoes an abrupt change (or step) at the cladding boundary. This is called a step-index fiber. In the graded-index fiber, the core refractive index varies as a function of the radial distance from the center of the fiber.

27 Optical Fibers for Biomedical Applications Fig. 4 Reflection and refraction of a light ray at a material boundary (After Keiser 2016)

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Normal line

n2 < n1

Refracted ray

θ2 φ2

Air Glass

φ1 n1

θ1 Incident ray

Material boundary

θ1 Reflected ray

Both the step-index and the graded-index fibers can be further divided into singlemode and multimode classes. As the name implies, a single-mode fiber (SMF) has only one propagating mode, whereas a multimode fiber (MMF) contains many hundreds of modes. Figure 3 gives examples of a few representative sizes of singleand multimode fibers to provide an idea of the dimensional scale. Multimode fibers have several advantages compared with single-mode fibers. First, the larger core radii of multimode fibers make it easier to launch optical power into the fiber and to collect light emitted or reflected from a biological sample. An advantage of multimode graded-index fibers is that they have larger data rate transmission capabilities than a comparably sized multimode step-index fiber. Single-mode fibers are more advantageous when delivering a narrow light beam to a specific tissue area and also are needed for applications that deal with coherence effects between propagating light beams. In standard conventional fibers, the core of radius a has a refractive index n1 , which for silica-based fibers is typically equal to 1.48. The core is encapsulated by a cladding of slightly lower index n2 , where n2 D n1 (1  ). The parameter  is the core-cladding index difference or simply the index difference. Typical values of  range from 1% to 3% for multimode fibers and from 0.2% to 1.0% for single-mode fibers.

Ray Optics Concepts For a conceptual illustration of how light travels along a fiber, consider the situation when the core diameter is much larger than the wavelength of the light. In this picture a basic geometric optics approach based on the concept of light rays can be used. To start, first consider Snell’s law, which describes what happens when a light ray is incident on the interface between two different materials. When a light ray encounters a smooth interface that separates two different dielectric media, part of the light is reflected back into the first medium, and the remainder is bent (or refracted) as it enters the second material. This is illustrated in Fig. 4 for the interface between two materials with refractive indices n1 and n2 , where n2 < n1 .

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Fig. 5 Ray optics picture of the propagation mechanism in an optical fiber (After Keiser 2016)

Propagating rays are captured in the acceptance cone

n < n1

Refracted ray lost in the cladding

n2 Cladding θ

θ0

f

Reflected ray

n1 Core

Non-captured rays outside of the acceptance cone

n2 Cladding

The relationship between the incident, reflected, and refracted rays at the interface is known as Snell’s law and is given by n1 sin ™1 D n2 sin ™2

(1)

n1 cos ®1 D n2 cos ®2

(2)

or, equivalently, as

where the angles are defined in Fig. 4. The angle ™1 between the incident ray and the normal to the surface is known as the angle of incidence. The application of Snell’s law to an optical fiber can be illustrated by means of Fig. 5. Here a light ray enters a fiber core from a medium of refractive index n at an angle ™0 with respect to the fiber axis. Inside the fiber the light ray strikes the corecladding interface at an angle ¥ relative to the normal of the interface. If this angle is such that the light ray is totally internally reflected at the core-cladding interface, then the ray follows a zigzag path along the fiber core. Total internal reflection occurs when the refracted angle is ™2 D 90ı . Thus, from Snell’s law, the minimum or critical angle ¥c that supports total internal reflection at the core-cladding interface is given by sin ¥c D

n2 n1

(3)

Incident rays meeting the interface at angles less than ¥c will refract out of the core and be lost in the cladding, as the dashed line in Fig. 5 shows. By applying Snell’s law to the air-fiber face boundary, the condition of Eq. 3 can be related to the maximum entrance angle ™0, max , which is called the acceptance angle ™A , through the relationship: 1=2  nsin™0;max D nsin™A D n1 sin™c D n21  n22

(4)

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where ™c D  /2 – ¥c . Those rays having entrance angles ™0 less than ™A will be totally internally reflected at the core-cladding interface. Thus, ™A defines an acceptance cone for an optical fiber. Rays outside of the acceptance cone, such as the ray shown by the dashed line in Fig. 5, will refract out of the core and be lost in the cladding. Equation 4 also defines the numerical aperture (NA) of a step-index fiber: p 1=2  NA D nsin™A D n21  n22  n1 2

(5)

The approximation on the right-hand side of the above equation holds because the parameter  is much less than 1. Because it is related to the acceptance angle, the NA is used commonly to describe the light acceptance or gathering capability of a multimode fiber and to calculate the source-to-optical fiber power coupling efficiencies. The NA value can be found on vendor data sheets for fibers. As an example, consider a multimode step-index silica fiber that has a core refractive index n1 D 1.480 and a cladding index n2 D 1.460. From Eq. 3, the critical angle is given by ®c D sin1

n2 1:460 ı D 80:5 D sin1 n1 1:480

From Eq. 5 the numerical aperture is  1=2 NA D n21  n22 D 0:242 From Eq. 5 the acceptance angle in air (n D 1.00) is ™A D sin1 NA D sin1 0:242 D 14:0

ı

Modal Concepts The light ray picture gives a general concept of how light propagates along a fiber. However, mode theory is necessary to get a more detailed understanding of concepts such as mode coupling, dispersion, coherence or interference phenomena, and light propagation in single-mode and few-mode fibers. Figure 6 shows a longitudinal cross-sectional view of an optical fiber, which illustrates the field patterns of some of the lower-order transverse electric (TE) modes. The order of a mode is equal to the number of field zeros across the guiding core. The plots illustrate that the electric fields of the guided modes are not completely confined to the core but extend partially into the cladding. These cladding-mode components are important for understanding the operation of certain types of optical fiber sensors. Inside the core region of refractive index n1 , the fields vary harmonically, and they decay exponentially in the cladding of refractive index n2 . The exponentially decaying

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Evanescent tails of the modes extend into the cladding

Cladding n2

Zero crossings

Core n1

Cladding n2 Fundamental LP01mode

Harmonic variation in the core

Exponential decay in the cladding Second-order Third-order mode mode

Fig. 6 Electric field distributions of lower-order guided modes in an optical fiber (After Keiser 2016)

field is referred to as an evanescent field. For low-order modes, the fields are tightly concentrated near the optical fiber axis and penetrate very little into the cladding region. Higher-order mode fields are distributed more toward the edges of the core and penetrate farther into the cladding. As the optical fiber core radius is made progressively smaller, all modes except the fundamental mode (the zeroth-order linearly polarized mode designated by LP01 ) shown in Fig. 6 will start getting cut off and will not propagate in the fiber. A single-mode fiber results when only the fundamental mode can propagate along the fiber axis. An important parameter related to the cutoff condition is the V number, which is defined by VD

1=2 2 a p 2 a  2 2 a n1  n22 NA  n1 2 D œ œ œ

(6)

where the approximation on the right-hand side comes from Eq. 5. The V number is a dimensionless parameter that determines how many modes a fiber can support. Except for the lowest-order fundamental mode, each mode can exist only for values of V that exceed the limiting value V D 2.405 (with each mode having a different V limit). The wavelength at which all higher-order modes are cut off is called the cutoff wavelength œc . The fundamental mode has no cutoff and ceases to exist only when the core diameter is zero. The V number also can be used to express how many modes M are allowed in a multimode step-index fiber when V is large. For this case, an estimate of the total number of modes supported in such a fiber is    1 2 a 2  2 V2 MD n1  n22 D 2 œ 2

(7)

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Because the field of a guided mode extends partly into the cladding, as shown in Fig. 6, another important parameter is the fractional power flow in the core and cladding for a given mode. As the V number approaches cutoff for any particular mode, more of the power of that mode is in the cladding. At the cutoff point, all the optical power of the mode resides in the cladding. For large values of V that are far from the cutoff condition, the fractional optical power residing in the cladding can be estimated by Pclad 4  p P 3 M

(8)

where P is the total optical power in the fiber.

Graded-Index Optical Fibers Core Index Structure In contrast to a step-index fiber in which the core index is constant, in a gradedindex fiber, the refractive index of the core decreases with increasing radial distance r from the fiber axis and is constant in the cladding. A commonly used refractiveindex variation is the power law relationship: h  r ’ i1=2 n .r/ D n1 1  2 for 0  r  a a

(9)

D n1 .1  2/1=2  n1 .1  / D n2 for r  a Here, r is the radial distance from the fiber axis, a is the core radius, n1 is the refractive index at the core axis, n2 is the refractive index of the cladding, and the dimensionless parameter ’ defines the shape of the index profile. For example, a value of ’ D 2.0 describes a parabolic index profile, whereas when ’ D 1 the fiber has a step-index profile. The index difference  for the graded-index fiber is given by D

n21  n22 n1  n2  2 n1 2n1

(10)

The approximation on the right-hand side reduces this expression for  to that of the step-index fiber. Thus, the same symbol  is used in both cases.

Graded-Index Numerical Aperture Whereas for a step-index fiber the NA is constant across the core, for graded-index fibers, the NA is a function of position across the core end face. Geometrical optics analyses show that light incident on the fiber core at position r will propagate as a

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guided mode only if it is within the local numerical aperture NA(r) at that point. The local numerical aperture is defined as p  1=2 NA .r/ D n2 .r/  n22  NA.0/ 1  .r=a/’ for r  a D 0 for r > a

(11)

where the axial numerical aperture is defined as p 1=2  2 1=2  D n1  n22  n1 2 NA.0/ D n2 .0/  n22

(12)

Thus, the NA of a graded-index fiber decreases from NA(0) to zero as r moves from the fiber axis to the core-cladding boundary. The number of bound modes Mg in a graded-index fiber is   2 a 2 2 ’ ’ V2 Mg D n1   ’C2 œ ’C2 2

(13)

Typically, a parabolic refractive index profile given by ’ D 2.0 is used for a graded-index fiber. In this case, the number of modes is Mg D V2 /4, which is half the number of modes supported by a step-index fiber (for which ’ D 1) that has the same V value.

Cutoff Wavelength in Graded-Index Fibers Similar to step-index fibers, graded-index fibers can be designed as single-mode fibers in which only the fundamental mode propagates at a specific operational wavelength. An empirical expression of the V number at which the second lowestorder mode is cut off for graded-index fibers has been shown to be r Vcutoff D 2:405 1 C

2 ’

(14)

Equation 14 shows that in general for a graded-index fiber, the value of Vcutoff decreases as the profile parameter ’ increases. This also shows that in a parabolic graded-index fibers (’ D 2), the critical value of V for the cutoff condition is a factor p of 2 larger than for a similar-sized step-index fiber. In addition, from the definition of V given by Eq. 6, the numerical aperture of a graded-index fiber is larger than that of a step-index fiber of comparable size.

Performance Characteristics of Generic Optical Fibers When selecting an optical fiber to use in a particular biophotonics system application, various performance characteristics need to be considered. These include optical signal attenuation as a function of wavelength, optical power-handling

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capability, the degree of signal loss as the fiber is bent, and mechanical properties of the optical fiber.

Attenuation Versus Wavelength Attenuation (i.e., decrease in optical power) in a fiber is due to absorption, scattering, and radiative losses of optical energy as light propagates along a fiber. This parameter normally is measured in units of decibels per kilometer (dB/km) or decibels per meter (dB/m). A variety of materials are used to make different types of optical fibers for biophotonic applications. The basic reason for this selection is that each material type exhibits different light-attenuation characteristics in various spectral bands. For example, silica (SiO2 ) glass is the common material used for solid-core fibers in telecom applications. This material has low losses in the 800– 1600 nm telecom operating region, but the loss is significantly higher for ultraviolet and mid-infrared wavelengths. Thus, other fiber types and/or materials are needed for biophotonic applications that use wavelengths outside of the telecom spectral band. Bend-Loss Insensitivity When conventional and certain other types of optical fibers are bent, light escapes from the core of the curved fiber section. This bending loss increases exponentially as the bending radius decreases. The loss is unobservable for slight bends, but at a certain critical radius, the loss becomes observable. If the bend radius is made smaller than the critical radius, the bend losses quickly become extremely large. Specially designed fibers that are less sensitive to bending loss are available commercially to provide optimum low bending loss performance at specific operating wavelengths, such as 820 or 1550 nm. These fibers usually have an 80 m cladding diameter. Such a smaller outer diameter yields a reduced coil volume compared with a standard 125 m cladding diameter when a length of this low-bend-loss fiber is coiled up within a miniature optoelectronic device package or in a compact biophotonics instrument. Mechanical Properties Several unique mechanical properties of optical fibers make them appealing for biomedical applications. For example, the fact that optical fibers are a thin highly flexible medium allows minimally invasive medical treatment or diagnostic procedures to take place in a living body. Such applications include endoscopic procedures, cardiovascular surgery, and microsurgery. Another mechanical-related characteristic is that by monitoring the signal change resulting from some intrinsic physical variation of an optical fiber (e.g., elongation or refractive index changes), one can create fiber sensors to measure external physical parameter changes. For example, if a varying external parameter, such as temperature or pressure fluctuations, elongates the fiber or induces refractive index difference changes at the outer cladding boundary, this effect can modulate the intensity, phase, polarization, wavelength, or transit time of light in the fiber.

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The degree of light modulation then can be used to directly measure changes in the external physical parameter. For biophotonic applications, the external physical parameters of interest include fluctuations in pressure, temperature, stress and strain in tissue and bones, and the molecular composition of a liquid or gas surrounding the fiber.

Optical Power-Handling Capability In biomedical photonics applications such as imaging and fluorescence spectroscopy, the fibers transmit optical power levels of less than 1 W. In other applications the fibers need to transmit optical power levels of 10 W and higher. A principal application is laser surgery, which includes bone ablation, cardiovascular procedures, dentistry, dermatology, ophthalmology, and oncological treatments. Optical fibers that can transmit high-power levels include hard-clad silica fibers with fused silica cores that contain very low contaminant levels, coated hollow-core fibers, photonic crystal fibers, and germinate glass fibers.

Optical Fibers Used in Biophotonics This section describes various categories of optical fiber structures that are appropriate for use in different biophotonic spectral bands. The fiber categories include conventional and specialty multimode and single-mode solid-core fibers, doubleclad fibers, hard-clad silica fibers, conventional coated hollow-core fibers, photonic crystal fibers, polymer optical fibers, side-emitting and side-firing fibers, middleinfrared fibers, and optical fiber bundles.

Conventional Solid-Core Fibers Extensive development work by telecom companies has resulted in highly reliable and widely available conventional solid-core silica-based optical fibers. These fibers come in a variety of core sizes and are used throughout the world in telecom networks and also in many biophotonic applications. Figure 7 shows the optical signal attenuation per kilometer as a function of wavelength. The shape of the attenuation curve is due to the following three factors: (a) Intrinsic absorption resulting from electronic absorption bands causes high attenuations in the ultraviolet region for wavelengths less than about 500 nm. (b) The Rayleigh scattering effect starts to dominate the attenuation for wavelengths above 500 nm. However, this effect decreases rapidly with increasing wavelength because of its 1/œ4 behavior. (c) Atomic vibration bands in the optical fiber material produce intrinsic absorption in the infrared spectrum. This is the dominant attenuation mechanism in the infrared region for wavelengths above 1500 nm.

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100 Infrared absorption loss Measured fiber loss

Attenuation (dB/km)

10

1 Ultraviolet absorption loss

0.1

Scattering loss

0.01 0.5

0.6

0.8 1 1.2 1.5 2 Wavelength (μm)

5

Fig. 7 Typical attenuation curve of a silica fiber as a function of wavelength (After Keiser et al. 2014)

Figure 7 illustrates that in silica fibers, these attenuation mechanisms result in a low-loss region in the spectral range of 700–1600 nm, which matches the lowabsorption biophotonics window illustrated in Fig. 1. Absorption by residual water ions in the silica material gives rise to an attenuation spike around 1400 nm. Removing a large percentage of the water ions during fiber manufacturing can reduce this attenuation spike significantly. The end product is called a low-water-content fiber or low-water-peak fiber. Multimode fibers are commercially available with standard core diameters of 50, 62.5, 100, 200 m, or larger. Applications of such multimode fibers include laser or LED light delivery, photobiomodulation, optical fiber probes, and oncological therapy. Single-mode fibers have core diameters around 10 m and are used in clinical fiber sensors, in endoscopes or catheters, and in imaging systems.

Specialty Solid-Core Fibers Specialty solid-core fibers can be custom-designed through either material or structural variations. Such fibers enable functions such as lightwave signal manipulation for optical signal-processing functions, extending the spectral operating

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Intensity

Input spectrum Transmission spectrum

Cladding Wavelength

Core Bragg grating typically a few mm or cm

Reflection spectrum (λB) Fig. 8 A periodic index variation in the core of a single-mode fiber creates a fiber Bragg grating (After Keiser et al. 2014)

range of the fiber, sensing fluctuations in a physical parameter such as temperature or pressure, or analyzing the contents of biomedical fluids. The main specialty solid-core fiber types for biophotonic applications are photosensitive fibers, fibers resistant to darkening from ultraviolet light, bend-loss-insensitive fibers for use along circuitous paths inside bodies, and polarization-preserving optical fibers for imaging. Descriptions and utilizations of such fibers are given in the following subsections.

Photosensitive Optical Fiber In a photosensitive fiber, the refractive index changes when the fiber material is exposed to ultraviolet light. This feature can be employed to fabricate a short fiber Bragg grating (FBG) in the fiber core, which is a periodic variation of the refractive index along the fiber axis (Kashyap 2010; Al-Fakih et al. 2012). This index variation is illustrated in Fig. 8, where n1 is the core refractive index, n2 is the cladding index, and ƒ is the period of the grating, that is, the spacing between the maxima of the index variations. If a specific incident lightwave at a wavelength œB (which is known as the Bragg wavelength) encounters a periodic refractive index variation, œB will be reflected back if the following condition is met: œB D 2neff ƒ

(15)

Here neff is the effective refractive index, which has a value falling between the refractive indices n1 of the core and n2 of the cladding. The grating reflects the Bragg wavelength and all others will pass through. Fiber Bragg gratings are available in a selection of Bragg wavelengths with the width of the reflection bands at specific wavelengths varying from a few picometers to tens of nanometers. A common biophotonics utilization of a FBG is to sense a variation in a physical parameter such as temperature or pressure. For example, an external strain will slightly stretch the fiber, which causes the period ƒ of the FBG to lengthen and

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thus will change the characteristic Bragg wavelength value. Similarly, rises or drops in temperature will lengthen or shorten the value of ƒ. The resuting change in the Bragg wavelength B then can be related to the change in temperature.

Fibers Resistant to UV-Induced Darkening Conventional solid-core silica fibers darken when exposed to ultraviolet (UV) light. The UV-induced darkening losses are known as solarization and occur strongly at wavelengths less than about 260 nm. Consequently, standard silica optical fibers can only be implemented above approximately 300 nm. However, recently special glass processing techniques have allowed the fabrication of fibers with moderate (around 50% attenuation) to minimal (a few percent additional loss) UV sensitivity below 260 nm (Khalilov et al. 2014; Gebert et al. 2014). When these solarization-resistant fibers are exposed to UV light, the transmittance decreases rapidly and then stabilizes to an asymptotic value. The exact asymptotic value and the time duration needed to reach the plateau depend on the specific fiber type and on the UV wavelength. Shorter wavelengths induce larger attenuation changes, and a longer time is needed to reach the asymptotic value. Solarization-resistant fibers for use in the 180–850 nm range are commercially available with core diameters ranging from 50 to 1000 m and a numerical aperture of 0.22. Manufacturers recommend that prior to use, these fibers should first be exposed to UV radiation for approximately 5 min or more (depending on the operational wavelength) to establish loss equilibrium.

Bend-Insensitive Fiber Often when employing optical fibers within a living body for medical applications, the fibers follow a winding path with sharp bends through arteries that snake around bones and organs. As noted in section “Bend-Loss Insensitivity,” radiative losses can occur whenever the fiber undergoes a bend with a finite radius of curvature. This factor is negligible for slight bends. However, the bending loss effects increase exponentially as the radius of curvature decreases until at a certain critical radius the losses become extremely large. The bending loss is more sensitive at longer wavelengths as Fig. 9 shows. For example, suppose a conventional fiber has a 1 cm bending radius as indicated by the dashed vertical line in Fig. 9. At 1310 nm this results in an additional loss of 1 dB. However, for this 1 cm bend radius, there will be an additional loss of about 100 dB at 1550 nm. Optical fiber research activities in the telecom industry led to bend-lossinsensitive fibers that can tolerate numerous sharp bends. These same fibers also can be applied in the medical field (Matsui et al. 2011; Kusakari et al. 2013). The fibers are available with either an 80 m or a 125 m cladding diameter as standard products. The 80 m reduced-cladding fiber results in a much smaller volume compared with a 125 m cladding diameter when a fiber length is coiled up within a miniature optoelectronic device package or in a compact biophotonics instrument.

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Fig. 9 Generic bend-loss sensitivity for conventional fibers at 1310 and 1550 nm (After Keiser 2016)

Bending loss at 1550 nm

Bending loss (dB)

1000

100

100 dB bending loss at 1550 nm

Bending loss at 1310 nm

10 1 dB bending loss at 1310 nm

1

0.1 0.5

1.0 Bending radius (cm)

1.5

y

y

Cladding

Cladding

Core

Core

x

x nx Horizontal mode sees an effective index nx

ny

Vertical mode sees an effective index ny

Fig. 10 Two polarization states of the fundamental mode in a single-mode fiber (After Keiser 2016)

Polarization-Maintaining Fibers The fundamental mode in a single-mode fiber can be viewed as consisting of two orthogonal polarization modes. These modes can be designated as horizontal and vertical polarizations in the x direction and y direction, respectively, as shown in Fig. 10. In general, the electromagnetic field of the light traveling along the fiber is a linear superposition of these two orthogonal modes and depends on the polarization state of the light at the input point of the fiber. In ideal fiber with perfect rotational symmetry, the polarization state of any light injected into the fiber will propagate unchanged along the fiber. In actual fibers any small imperfections (such as asymmetric lateral stresses, noncircular cores, and variations in refractive-index profiles) disturb the circular symmetry of the ideal fiber. The two orthogonal polarization modes then travel along the fiber with

27 Optical Fibers for Biomedical Applications

Core

Cladding

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Core

Stress applying elements PANDA structure

Bowtie structure

Fig. 11 Cross-sectional geometry of two different polarization-maintaining fibers (After Keiser 2016)

different phase velocities. This effect causes the state of polarization to fluctuate as a lightwave travels through the fiber. Specially designed polarization-maintaining fibers have been created that preserve the state of polarization along the fiber with little or no coupling between the two modes. Figure 11 illustrates the cross-sectional geometry of two commonly used polarization-maintaining fibers. The light circles represent the usual core and the cladding material. The dark areas are stress elements made from a different type of glass that are embedded in the cladding. The purpose of the stress-applying components is to create slow and fast axes in the core that will guide light in each mode at a different velocity. Thereby when polarized light is launched into the fiber it will maintain its state of polarization as it travels along the fiber. These fibers are used in special biophotonic applications such as fiber-optic sensing and interferometry where polarization preservation is essential (Tuchin et al. 2006).

Double-Clad Fibers A double-clad fiber (DCF) is being used widely in the medical field for imaging systems (Lemire-Renaud et al. 2011; Liang et al. 2012; Beaudette et al. 2015). As shown in Fig. 12, a DCF structure contains a core region, an inner cladding, and an outer cladding arranged concentrically. Typical dimensions are a 9 m core diameter, a 105 m inner cladding diameter, and an outer cladding with a 125 m diameter. Light in the core region is single-mode and is multimode in the inner cladding. The combination of a single-mode and multimode structure allows using just one optical fiber for the delivery and collection of probing light. The illumination light is transmitted via the single-mode core, and the multimode inner cladding is employed for the collection of light coming from the tissue.

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G. Keiser

Refractive index Core

Radius Inner cladding Outer cladding Buffer jacket Fig. 12 Cross-sectional representation of a DCF and its index profile (After Keiser 2016)

Table 2 General specifications of selected HCS fibers Core diameter Cladding diameter Wavelength range (high OH content) Wavelength range (low OH content) Maximum power capability (CW) Maximum power capability (pulsed) Maximum long-term bend radius Max attenuation at 850 nm

HCS fiber 1 200 ˙ 5 m 225 ˙ 5 m 300–1200 nm

HCS fiber 2 600 ˙ 10 m 630 ˙ 10 m 300–1200 nm

HCS fiber 3 1500 ˙ 30 m 1550 ˙ 30 m 300–1200 nm

400–2200 nm

400–2200 nm

400–2200 nm

0.2 kW

1.8 kW

11.3 kW

1.0 MW

9.0 MW

56.6 MW

40 mm

60 mm

150 mm

10 dB/km (0.010 dB/m)

12 dB/km (0.012 dB/m)

18 dB/km (0.018 dB/m)

Hard-Clad Silica Fibers A multimode hard-clad silica (HCS) optical fiber is useful in applications such as laser light delivery, endoscopic procedures, oncological therapy, and biosensing systems. The HCS structure consists of a silica glass core that is encapsulated by a thin plastic cladding. The hard cladding gives greater fiber strength and reduces static fatigue effects in humid environments. Other features of these HCS fibers include bend insensitivity, long-term reliability, ease of handling, and resistance to harsh chemicals. Core diameters of commercially available HCS fibers range from 200 to 1500 m. Table 2 lists some performance parameters of three selected HCS fibers.

27 Optical Fibers for Biomedical Applications

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A common HCS fiber structure for medical applications comprises a 200 m core diameter and a cladding diameter of 230 m. Such fibers are very strong and have a low attenuation ( 0 (silica), a binary composition exists where there can be significant cancelation of the photoelastic constant (Ballato and Dragic 2013; Dragic and Ballato 2014). Figure 2 provides a representative comparison of selected materials systems, whereby marked reductions in gB are realized through this materials approach. Table 1 provides a more thorough compilation of the materials and their deduced Brillouin-related properties; note the richness of compounds that possess negative p12 values (Dragic et al. 2013a, b, 2010, 2014; Mangognia et al. 2013; Cavillon et al. 2016). Further inspection of Fig. 2 shows that, while there is only one composition for a given system whereby gB D 0, the binaries have wide compositional ranges where gB is reduced by 10 dB or more.

Stimulated Raman Scattering (SRS) SRS is an interaction between an optical wave and optical phonons and can be considered a parasitic effect in high-peak-power fiber laser systems where wavelength control is mandatory (such as in spectrally beam-combined systems).

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Fig. 2 Comparison of relative Brillouin gain from fibers fabricated using the molten core method; see Table 1 for additional details

Table 1 Deduced Brillouin-related physical parameters for oxide compounds SiO2 GeO2 P2 O5 B2 O3 Y3 Al5 O12 Al2 O3 MgO SrO BaO Yb2 O3 La2 O3 Lu2 O3 Li2 O a b

Va (m/s) 5970 3510 3936 3315 7649 9790 8731 3785 3131 4110 3979 3660 6500

Not yet characterized nref D 11 GHz

 (kg/m3 ) 2200 3650 2390 1820 3848 3350 3322 4015 4688 8102 5676 7928 3150

B b (MHZ) 21 124 177 428 253 274 a

187 178 1375 181 145 a

n 1.444 1.571 1.488 1.41 1.868 1.653 1.81 1.81 1.792 1.881 1.877 1.66 1.97

p12 0.226 0.268 0.255 0.298 0.022 0.027 a

0.245 0.33 0.123 0.027 0.043 0.01

References Dragic et al. (2013c) Dragic and Ward (2010) Law et al. (2012) Dragic (2011) Dragic et al. (2010) Dragic et al. (2013c) Mangognia et al. (2013) Cavillon et al. (2016) Dragic et al. (2013b) Dragic et al. (2013c) Dragic et al. (2014) Dragic et al. (2016a) Dragic et al. (2015)

33 Materials Development for Advanced Optical Fiber Sensors and Lasers

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The excitation of SRS from spontaneous scattering can lead to wavelength shifts and power instabilities that degrade laser performance, thus presenting the need for its suppression. However, unlike Brillouin scattering, the dependence of the Raman gain coefficient (gR ) is on material properties (Raman cross section, refractive index) that do not permit a zeroing of its value. Instead, an intrinsically low Raman gain material must be selected such that one or more of the following conditions are met: (a) the material is highly disordered so as to broaden the Raman gain spectrum thereby reducing the peak value; (b) high concentrations of materials with low gR are utilized; and (c) materials are utilized that have Raman spectra with minimal overlap. For (c), a mixture of two materials with similar strength but not overlapping Raman gain spectra could cut the peak Raman gain coefficient in half (Dragic and Ballato 2013). In compositions possessing large yttria (Y2 O3 ) and alumina concentrations, significant Raman gain reduction can be realized (with 3 dB reduction relative to silica demonstrated experimentally in a fabricated fiber), without the use of complicated fiber filtering schemes. Figure 3 shows the relative gR for yttrioaluminosilicate glass versus Y2 O3 C Al2 O3 content (in mole%) (Dragic and Ballato 2013). These reductions only partly relate to the weaker gR in yttria and

Fig. 3 gR relative to pure silica for crystal derived optical fiber as a function of total non-silica content. The numbers denote specific fibers described in Dragic and Ballato (2013) and Ballato and Dragic (2014)

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alumina (which has replaced the much higher-gain silica). The reduction also is due, in part, to the highly non-equilibrium nature of the molten core method employed to fabricate these fibers. In this case, the glass is realized directly from the melt such that the distribution of molecules comprising the glass is more disordered, further lowering the peak gR values for each of the contributing vibrations.

Transverse Mode Instability (TMI) Thermally induced longitudinal modulations of the refractive index associated with stimulated Rayleigh scattering result in modal instabilities (mode coupling) in “effectively single-mode” fiber lasers. These higher-order transverse mode instabilities (TMIs) limit power scaling in fiber laser systems by dynamically randomizing the beam modal distribution. The link between TMI and stimulated Rayleigh scattering yields a materials solution to TMI. The TMI threshold is proportional to ¡C/Q where ¡ is the mass density, C is the specific heat, and Q is the thermo-optic coefficient (dn/dT) (Dong 2013; Smith and Smith 2016). Here, in an analogous manner to Brillouin scattering, a material with dn/dT D 0 would completely obviate TMI. Accordingly, combining materials with thermo-optic coefficients of opposite sign can give rise to a significant reduction in dn/dT and possibly even yielding a value of zero. Materials such as SiO2 , GeO2 (dn/dT larger than silica), and Al2 O3 (dn/dT similar to silica) have positive dn/dT, but this value is also negative for several materials such as P2 O5 and B2 O3 among many others. As an illustrative example based on B2 O3 doped optical fibers, the thermo-optic coefficient for borosilicate glass is shown in Fig. 4. The origins of positive and negative values of dn/dT are fully analogous to those for positive and negative p12 : a heated material thermally expands, and the resulting decrease in density leads to a decrease in refractive index via the Clausius– Mossotti relation. However, this expansion can also tend to increase the electronic polarizability (i.e., increase in refractive index) of the material. As such negative dn/dT materials are dominated by thermal expansion and positive dn/dT materials by the increase in polarizability. The additive materials model, discussed in detail below, was employed to generate the solid curve shown. n2 -Related Effects Relative to Brillouin and Raman gain suppression through the enabling materials, less work has been conducted on compositions to reduce the nonlinear refractive index, n2 . The general effect of n2 -related nonlinear optical processes is to broaden and modify the optical spectrum and is undesirable in high-peak-power laser systems (Ballato and Dragic, 2014). Accordingly, suppression of parametric nonlinear phenomena requires a minimization of n2 . It is further worth noting that the Raman gain is proportional to the imaginary part of (3) , and so low n2 materials should further aid in the reduction in SRS. One approach to n2 reduction would be to incorporate greater amounts of fluorine into the glass structure, since compounds containing fluorine are known to possess reduced n2 values relative to silica (Boling et al. 1978; Nakajima and

33 Materials Development for Advanced Optical Fiber Sensors and Lasers

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Fig. 4 Thermo-optic coefficient of borosilicate glass with increasing B2 O3 content. A measured datum on a fabricated fiber where the thermo-optic coefficient, dn/dT, is reduced by 3 dB is provided

Ohashi 2002). However, there are two principal issues with fluorination. First, due both to the glass structure and also to the very high processing temperatures associated with CVD methods, the incorporation of fluorine into silica (F:SiO2 ) is quite low, typically on the order of 1% (Schuster et al. 2014). While this is useful for conventional and selected double-clad fiber refractive index profiles, it does not afford much reduction in n2 . Secondly, while fluoride glasses are naturally more linear (lower nonlinearities) than oxides, they are not compatible with HEL demands. It would therefore be reasonable to focus such “intrinsically low-n2 ” glasses on refractory oxyfluoride compositions since, for practical reasons, HEL fibers must be silica based yet exhibit considerably higher fluorine levels than otherwise possible. As noted, the high temperatures (2400 ı C for final preform collapse) inherent in conventional CVD optical fiber processes facilitate evaporation of most fluorine doped into the glass. While the molten core process does necessarily imply that the core phase is molten, its temperature usually does not exceed about 2000 ı C, and fiber often can be achieved at temperatures closer to 1900 ı C such that more fluorine remains in the core glass. Further, high-melting point fluoride compounds with low vapor pressures that are combined with the intrinsically low Brillouin or Raman oxide compounds create a fascinating new class of oxyfluoride glasses with the potential for significant reductions in SBS, SRS, and n2 -related nonlinearities.

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Materials Modeling Parasitic nonlinear phenomena are light-matter interactions governed by the properties of the material in which they occur. Given the nearly boundless range of glass compositions that are possible, empirically isolating a particular material that is optimized for one or more applications is a tremendously daunting proposition. As such, modeling approaches are applied to experimental data in order to guide the selection of appropriate application-specific materials. Through a number of various oxide materials systems spanning a broad range of the periodic table (Ballato and Dragic 2013), models based on the Winkelmann–Schott (W–S) (Winkelmann and Schott 1894; Dragic and Ballato 2014; Ballato et al. 2016) method of addition (law of mixtures) have clearly been shown to be powerful tools that can be utilized compositionally to design glass fibers for a number of applications. This specifically includes those with suppressed parasitic phenomena, such as SBS and SRS, respectively, and TMI, and fibers with physical properties that are optimized for certain sensing applications. Specific achievements include prediction of a zeroSBS material and fibers whose Brillouin frequency shift is immune to changes in temperature, applicable to distributed sensing systems. This model has thus far only been applied to oxide glasses possessing at least some silica in their composition. Numerous additive models for the design of glass with specific desirable properties have been developed, and a very nice review has been provided by Volf (1988). However, despite a rich history of glass modeling efforts by those studying the nature and design of glass, very few materials models have been applied to the design of optical fibers that utilize multicomponent glasses. One very well-known example of such a model is a form of the Sellmeier equation for mixed glasses, such as GeO2 –SiO2 core glass, that can predict a refractive index as a function of composition (Fleming 1984). A key feature of this model type is the assumption that GeO2 and SiO2 are treated as being independent and that they together form a simple mixture. This is a reasonable expectation considering that these two compounds are network formers. Turning to a visual representation, Fig. 5 shows how a unit length of the core glass may be “unmixed,” giving rise to a region of pure SiO2 and one of pure GeO2 . Since the refractive index is a measure of the speed of light (phase velocity), Fig. 5 suggests a means to calculate the refractive index of the mixed glass: calculate the total time of flight (TOF) of a photon through the glass then divide by the vacuum speed of light. The total TOF is t D nc1 L1 C nc2 L2 where n is the refractive index, c is the vacuum speed of light, and L is the length of the segment. For a normalized unit length (L1 C L2 D 1) of material, the Ls represent the volume fraction of the material represented by a particular segment. This summation could be extended with additional terms for each new component added to the glass. As such, this can be mathematically represented by a finite sum of the product of the refractive index multiplied by volume fraction. In other words, the material refractive index is modeled as a volume average of the refractive indices of the constituents.

33 Materials Development for Advanced Optical Fiber Sensors and Lasers

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Fig. 5 A mixture representation of multicomponent glass. Top: a mixture of components gives rise to the aggregate glass. Bottom: aggregate glass separated into its pure constituents. If the segment is a normalized unit length (L1 C L2 D1), the Ls represent fractional volume of a constituent

This approach may be generalized to material properties other than the refractive index. Furthermore, it may also include network modifier constituents provided there is caution exercised in the interpretation of data. In short, starting with the fused SiO2 network, and its known bulk (and quenched) physical characteristics, material is added to the starting silica system via the following governing equation in its most basic form GD

XN iD1

gi x i

(2)

Here, x is the additivity parameter of constituent i, g is the physical property of that constituent, and G is the aggregate value. In the beginning example above, the refractive index was used as the physical property g and volume fraction the additivity parameter x. Taking the volume fraction as a more global x, gB (the Brillouin gain coefficient) may be calculated for an arbitrary composition if each of its variables are known, namely, density , refractive index n, photoelastic constant p12 , the acoustic velocity Va , and Brillouin spectral width ¤B (at some acoustic wave frequency ¤B ). Va normalized to some reference velocity is sometimes referred to as the acoustic index, and as with the refractive index, the acoustic index is the material property g which is to be added. In order to model the Pockels coefficients (p11 and p12 ), the g’s are derived from strain–stress relationships as (Ballato and Dragic 2013; Dragic and Ballato 2014; Ryan et al. 2015) C1;i D

1 3 1 ni Œp12;i  i .p11;i  p12;i / and C2;i D n3i Œp11;i  2i .p12;i / 2 2

(3)

where ¤i is the Poisson ratio for constituent i. With the assumption that the Poisson ratio adds (as g) according to Eq. 2, the p’s for the aggregate glass may be determined. More specifically, Eq. 3 becomes that of the aggregate glass if the subscript i is dropped. In that case Eq. 3 becomes a system of two equations and two unknowns (p11 and p12 for the aggregate glass).

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Equation 2 may be extended to include environmental conditions in a straightforward way. Each property that can be evaluated (e.g., Va or n) may be set to be functions of temperature or strain. For example, by making the replacement n ! no C (dn/dT)T, while recognizing that a change in temperature T is a constant for all constituents, the thermo-optic coefficient (dn/dT) may be calculated. Other glass characteristics that can be calculated include the coefficient of thermal expansion, change of index with strain or stress, and change of acoustic velocity with temperature or strain. However, in doing so additional precautions must be taken to account for the presence of a cladding. More details are provided in the next section. Such models clearly represent very powerful tools in the design of glasses that possess low nonlinearity, especially if one already knows how the addition of a constituent will change the aggregate physical property (see Table 1). If not, such models may be used “in the reverse” in order to determine empirical material values for the pure constituents. Working backward from experimental data, the g’s can be used as fit parameters, which would then subsequently be determined and then employed as design variables. Once again, in the case of “mixed” glasses that form networks (such as germanosilicates), the various g’s can be construed to be the bulk values of the individual constituents. For network modifiers (such as Group I or II oxides), they are likely more accurately interpreted to be their influence on the starting glass network, rather than some bulk value. In principle, this additivity model seems to apply well to isotropic glass systems but over limited experimentally characterized compositional ranges. Moreover, it is not even necessary for an ion (e.g., Al in Al2 O3 ) to have the same coordination, short-range order, etc. at every site. The additive model essentially provides an ensemble (“observed”) average (or net) effect for that glass component, and that can depend strongly on how and by whom the glass was fabricated. Clearly, this removes the use of the model to gain physical insight into the short-range order of glass systems, thus to be able to predict accurately how the glasses form from their precursors. In other words, neither the x’s nor g’s are predicted by the additive model but once determined still serve as a powerful engineering tool in the design of tailored optical fibers of added value. To understand and even predict glass formation (the x’s), an approach taken has been computational, based on a version of the Voigt–Reuss–Hill procedure (Ballato et al. 2016) to calculate “glassy” properties from those of the crystalline precursors. Excellent agreement was achieved between theory and experiment (i.e., those values determined using W–S methodologies) for alumina.

Modeling the Fiber Structure Optical fibers come in many flavors and many levels of complexity. The simplest fiber arguably is the two-layer core-cladding structure. Usually, the cladding is much larger, more voluminous than the core. Given that the core and cladding glasses can be very different both compositionally and physically, the composite-like nature of

33 Materials Development for Advanced Optical Fiber Sensors and Lasers

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Fig. 6 The effect of CTE mismatch. Left: an air-bound core glass will expand to a new radius with an increase in temperature. Right: a core clad in a low-CTE material will experience pressure with an increase in temperature due to its expansion being restricted. The core is essentially compressed from its unclad position in a thermally expanded state

the fiber system must be taken into consideration. Of notable importance is that the geometric configuration of the optical fiber can have a strong influence on the material properties of the glass comprising it, especially in the cases where the fiber environmental conditions might be changing. Along these lines, work has been done to reconcile a core-cladding mismatch in the coefficient of thermal expansion (CTE) (Dragic et al. 2016b). Take, for instance, a core glass (such as an aluminosilicate) possessing a CTE that is several times larger than that of the cladding (e.g., pure SiO2 , CTE  0.6  106 / ı C). A core that is bound entirely by air will expand at its rate of CTE with any increase in environmental temperature. This is illustrated in Fig. 6 in an end-view for a cylindrical glass rod (left image). However, the presence of a solid cladding (right image) will limit the extent of thermal expansion resulting in a net pressure on the core. This pressure can be construed to be a negative strain and therefore will influence any material properties that are dependent on it, such as refractive index and acoustic velocity. It is possible that an optical fiber has a composition that is changing in the radial direction (such as a graded-index fiber). In this case, any parameters extracted from the fiber via measurements will be those of the waveguide modes and not necessarily those of the materials. In a limit, should an acoustic or optical wave be confined tightly to the center of the core, the mode values will approximately be the material values in that region. If this is not the case, modeling the system then becomes complicated by the need to adjust material parameters throughout the core region while solving for mode values that best match the measured values. If the composition of the fiber is well-known, these requisite parameters can easily be calculated anywhere in the fiber utilizing Eq. 2. In this case the problem again reduces to one where the physical characteristics of the constituents can become the fitting parameters. Often, the determination of a fiber’s composition will be one providing only a few points across the core. In this case, the core may be partitioned into sections, corresponding to the available measurements. Then, each layer would possess a unique composition and therefore unique physical attributes. The example of a

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Fig. 7 Should a compositional analysis yield only a few data points (shown representatively in green), the core may be partitioned into segments. Each layer possesses a unique composition and therefore unique physical properties. A fictitious RIP is provided as an illustrative example. The RIP may be approximated by four layers (in red) in the core, corresponding to the four measured compositions. Solving for the mode is then a simple boundary condition problem

four-layer stepwise approximation to the refractive index profile (RIP) of a fiber is shown in Fig. 7. Outside of the outermost core layer is the cladding, which is typically a uniform material (such as pure silica). There are four boundaries in this example, and by enforcing the boundary conditions the modes can be calculated. A similar approach can be taken in characterizing the acoustic velocity relevant for Brillouin scattering. Importantly, the various layers will have material properties that are functions of temperature and/or strain, and therefore, the calculated modes will also have dependencies on these environmental conditions. This allows for the determination of material properties even from measurements on the modes of an optical fiber.

Applications of Intrinsically Low Nonlinearity Materials The applications of intrinsically low nonlinearity materials can be classified into two main categories: passive and active. The primary goal of passive optical fibers is to deliver or transmit information or power. Active fibers are those typically utilized in fiber lasers and amplifiers. Considering Raman and parametric amplifiers, active fibers need not be doped with a rare-earth element, but this discussion will be limited largely to the rare-earth-doped variety.

Passive Fibers The purpose of passive fibers in a light-wave system might be to deliver power, energy, or a signal to some other position or component in a system. Alternatively,

33 Materials Development for Advanced Optical Fiber Sensors and Lasers

1315

it may serve as the medium from which a thermomechanical environment is characterized (described in a later section). For example, it may serve the purpose to deliver high continuous-wave (CW) power to the bottom of a well that has been laser-drilled (Hecht 2012), deliver pulses for a lidar system remotely from the laser source, or even act as a simple interconnect between components in a high-power system. It is important to clearly identify the fiber length needed for a particular application, as this will ultimately limit the power handling capability (Smith 1972). To a good approximation, a so-called critical or threshold power can be identified for both the Brillouin and Raman scattering processes, defining a turn-on point SBS for the self-stimulated process. For the former Pcrit  21 gABeffL and for the latter A

SRS Pcrit  16 gReffL where gB and gR are the Brillouin and Raman gain coefficients (units of m/W), respectively, and Aeff is the effective mode area in the fiber. The effective area is defined to be

2 R1 R

!2 

rE .r; / E .r; / drd

Aeff D

0 0 2 R1 R

r.E

(4) .r; / E 

2

.r; // drd

0 0

where E is the electric field distribution. In the case where fiber attenuation is significant over the relevant lengths, L becomes an effective length defined as Leff D (1  exp(˛L))/˛ where ’ is the attenuation coefficient in the fiber (m1 ). In the case of pulses, the effective length to a reasonable approximation becomes (for Leff < L) Leff D ct/2nG where t is the pulse width and nG is the group index. For Brillouin scattering, this approximation breaks down for pulses that are comparable to, or shorter than, the acoustic damping or build-up times (usually 10’s of ns or less), and a more careful analysis is required. Finally, it should also be noted that these critical powers are more or less back-of-the-envelope estimates, which in practice serve as starting points in a design process. Ultimately, system requirements will drive power constraints. By way of example, consider a single-mode optical fiber at an operating wavelength that gives rise to a mode diameter of 10 m (Aeff D 78 m2 ). Using gB typical of Brillouin scattering in communications fibers (2.5  1011 m/W (Ballato and Dragic 2013)), one calculates a critical SBS threshold of 66 W-m. This suggests that in a 1-m length of fiber, 66 W of optical power may be transmitted prior to the onset of SBS. Presently, the most common method to enhance this threshold is through the broadening of the laser signal line width (where system constraints allow it). Also to good approximation, the Brillouin gain coefficient decreases by CL a factor B for a laser line width ( L ) that is broad relative to the Brillouin B line width ( B ). Consider, for example, the downhole application. Should the laser spectrum span 30 nm (9 THz) and assuming a  B of 70 MHz (a Brillouin gain spectrum for SMF-28 is provided in Fig. 8; the gain coefficient at the peak is found from Eq. 1), the SBS threshold may be increased to 8.5 MW-m. Therefore, one

Fig. 8 Brillouin gain spectrum for SMF-28 fiber normalized to gB and measured at 1064 nm. The small peaks on the higher-frequency side of the spectrum can be identified as higher-order acoustic modes (Koyamada et al. 2004)

P. Dragic and J. Ballato

Relative Gain Coefficient

1316

1.0 0.8 0.6 0.4 0.2 0.0 15.7

15.8

15.9

16.0

16.1

16.2

Frequency (GHz) 1.0

Relative Gain Coefficient

Fig. 9 Raman gain spectrum of SiO2 normalized to gR at the peak. The relative strength and widths of the various peaks give insight into the structure of the glass

0.8 0.6 0.4 0.2 0.0 200

400

600

800

1000

1200

Wave Number (cm–1)

could expect to be able to transmit on the order of 8.5 kW in a 1-km effective length. In order to increase the length and power, the fiber core could be made to be larger (i.e., multimode), and lengthwise stress or thermal gradients may also be introduced. However, the effectiveness of the latter methods diminishes when the laser spectrum is broad. This sheds light on the value and relevance of an intrinsically low Brillouin gain coefficient fiber, assuming the losses can be made to be low enough for this application. The Raman gain coefficient is on the order of 1  1013 m/W for typical mostly silica optical fibers. A Raman gain spectrum typical of such fibers is provided in Fig. 9. This puts the SRS threshold at roughly 200 higher than that of SBS (assuming no suppression methods have been utilized). In conventional silica fiber, the Raman gain bandwidth is broad and therefore influences both narrowand broadband light sources. Due to the large wavelength shift, state-of-the-art techniques to suppress SRS include introducing a distributed loss to the Ramanscattered signal (Taru et al. 2007). This methodology does enable several dB of suppression of SRS relative to conventional fibers, likely comparable to what could be achieved with materials engineering alone. The key advantage of the latter is the potential for a much simpler fiber structure.

33 Materials Development for Advanced Optical Fiber Sensors and Lasers 1.0 0.8 Relative Signal

Fig. 10 Normalized spectrum of a 100-ns 1-kW-peak-power Gaussian pulse through a single-mode fiber at a wavelength of 1 m of lengths 0.1 m (black line), 5 m (blue line), and 25 m (red line) illustrating the broadening due to SPM

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0.6 0.4 0.2 0.0 –100

–50

0 Frequency (MHz)

50

100

The main n2 -related parasitic phenomenon manifests as self-phase modulation (SPM) in pulsed systems. Since the refractive index is a function of the intensity through n2 , the time-varying intensity (i.e., pulse shape) gives rise to a time-varying refractive index that modulates the phase of the optical signal. This leads to a broadening of the laser spectrum, which, depending on the application, may be fully tolerable, partly tolerable (e.g., where it might be used to suppress other parasitics such as SBS), or not at all tolerable (such as where a spectrum must be strictly maintained). To illustrate this, Fig. 10 shows the spectrum of a 100 ns, 1-kWpeak-power Gaussian pulse propagating through different lengths of a conventional single-mode fiber. The spectral distortion is very apparent but only tolerable if the system application allows for it. The nonlinear refractive index for conventional single-mode fibers is on the order of 2  1020 m2 /W – 3  1020 m2 /W (Boskovic et al. 1996). The nonlinear phase change can be determined from D 2n2 /o Aeff (Agrawal 1995), which has units of rad/W-m. For the single-mode fiber example above, this gives rise to a value of about 2  103 rad/W-m. As a general guideline, the nonlinear phase change should be much less than about 2  to avoid large spectral broadening. For the example provided here, this translates into a practical threshold on the order of 3 kW-m. While Aeff can certainly be increased in order to decrease the effects of SPM, n2 may also be reduced through the judicious use of materials. As an example, the calculation of Fig. 10 is repeated in Fig. 11 for n2 that is ½ of the original value. Spectral broadening is essentially cut to ½ the extent.

Active Fibers The dominant application for rare-earth-doped fibers is fiber lasers. These fibers are of considerable interest for a myriad of applications because they are capable of producing multiple kilowatts of continuous-wave (CW) power in a single welldefined beam. However, due to the parasitic phenomena described above, namely, SBS, narrow line-width CW power has been limited to below about 1 kW (Zervas and Codemard 2014). SRS does not yet currently represent a fundamental limit to

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Fig. 11 Same calculation as Fig. 10 but with n2 taking ½ the value

1.0

Relative Signal

0.8 0.6 0.4 0.2 0.0 –100

–50

0 Frequency (MHz)

50

100

narrow line-width CW systems but may as laser power scales to beyond 10 kW (Dawson et al. 2008). In pulsed mode, especially as the pulse length is very long compared with the fiber length, these processes can limit peak power to values comparable to the CW limit. Methods to suppress these parasitics generally have involved microstructuring of the fiber in order to achieve effectively single-mode performance while expanding the mode field diameter. The other methodologies for the suppression of nonlinear phenomena described in the previous section (such as thermal or strain distributions) can also be applied. Each of these methods ultimately is limited by the complexities surrounding their practical implementation (e.g., fiber fabrication complexity, unstable mechanics, cost, etc.) (Ballato and Dragic 2013). In this section, two interesting applications for fiber lasers, particularly narrow line-width ones, are briefly described in the context of the relevant power limiters. These applications are lidar and high-energy lasers.

Lidar The field of lidar (light detection and ranging) is a rich and constantly evolving one with a history that predates the invention of the laser. There are a wide variety of lidar types (Fujii and Fukuchi 2005), including those based on both elastic and inelastic scattering. Applications span from simple range finders to wind speed sensors to systems that probe the composition of the ionosphere. Depending on the data that one wishes to retrieve, laser requirements can vary widely. Indeed, there currently is no single laser that can be used for all lidar applications. For instance, a range finder utilizing a laser operating at 1550 nm will not necessarily be able to detect the presence and quantity of CO2 in that same wavelength range, despite the presence of absorption features there. As such, much of the area of lidar involves the development of a singular laser type, designed and optimized for one, very focused application, and often is one that pushes the technological limitations often associated with that laser type. Fiber-based lasers are a particularly attractive light source for lidar due to several key features they possess: (1) they can be compact and lightweight and have (2) high

33 Materials Development for Advanced Optical Fiber Sensors and Lasers

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Fig. 12 Simple block diagram of the key elements of a pulsed monostatic lidar

efficiency, (3) tunability, and (4) access to wide wavelength ranges (including postprocesses such as frequency conversion or Raman shifting). Features 1 and 2 enable portable and space-based systems, while features 3 and 4 suggest a wide application space. However, despite these apparent advantages that fiber laser technologies may possess over other types, they have not penetrated into areas that would seem to be natural for these laser types. It is the aforementioned parasitic nonlinearities that have been described above that have limited the reach of fiber laser sources for a broader range of lidar applications. Figure 12 shows a very basic block diagram of a pulsed monostatic lidar. The laser beam is shown exiting through the center of the receiver aperture (“telescope”), although it could emerge from another position from within or outside that area. The laser beam has an associated divergence, which is usually further enhanced via propagation through a turbulent medium such as the atmosphere. Backscattered signal can be provided by any number of processes including Rayleigh and Mie (including aerosol) scattering, Raman scattering, and resonance fluorescence (RF). In RF lidars, a photon is absorbed and subsequently is reradiated into 4  steradians as spontaneous emission. In a pulsed system, the temporal resolution can be limited either by (1) the pulse length of the laser signal or (2) the integration time at the receiver. Since a photon must propagate a round trip through a scattering volume in one integration time (t, or pulse width whichever is larger), the range bin is given by z D ct/2. The total received backscattered signal (typically measured in photon counts) as a function of range z is given by the lidar equation, which is described most generally as (Wandinger 2005) 2 3 Zz c P .z/ D P0 AG.z/ˇ.z/ exp 42 ˛.z/dz5 2

(5)

0

where P0 is the transmitted power, £ is the pulse width, A is the receive area, and  represents all basic system-level contributions to receiver efficiency (e.g.,

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receiver quantum efficiency, transmittance of receiver optics, etc.). Furthermore, G(z) represents a geometric factor between the scattered signal and the receiver field of view. It can be expressed generally as G(z) D  (z)/z2 where ¦(z) is an overlap factor between the laser beam and receiver field of view. The z2 comes from the decreasing intensity the further from the scatterer. Finally ˇ(z) is the backscatter coefficient, which depends on the type or combination of scattering contributions, and ˛(z) the attenuation coefficient (m1 ) of the medium, with the factor of 2 once again accounting for a round trip. Equation 5 suggests multiple means to increase signal in a lidar system. Since those quantities related to the atmosphere are usually fixed, an increase in laser power, receiver area, or range bin seem like attractive possibilities. Ultimately, the measurement sensitivity of a system is a function of the system signal-to-noise ratio (SNR), and for laser remote sensing systems, this is often at the shot noise limit. In other words, the SNR is proportional to the square root of the received signal. Hence, a twofold increase in SNR requires a fourfold increase in laser power or receiver area. The latter may not be practical, and the former may be limited by the capabilities of the laser itself. For a fiber laser, these can be clearly identified given the constraints of a lidar system. These are identified below: Wavelength Fiber-based light sources have a wide wavelength reach. Fiber lasers in the master-oscillator power amplifier (MOPA) configuration can be seeded by semiconductor lasers that are wavelength stabilized and appropriate for many spectroscopic lidars. These include RF and differential absorption lidar (DIAL). Tunability In the latter scheme above, differential absorption lidar (DIAL), the laser wavelength is tuned from on to off of an absorption peak of some species (e.g., H2 O or CO2 in the atmosphere). This requirement is typically from the range of a few to 10’s of GHz. Fiber-based lasers, therefore, are finely suited for this given their wide-gain bandwidth and the fact that the seed laser diodes may be internally or externally wavelength modulated. Spectral Width and Pulsed Mode In many applications, a very narrow spectral width is required of the laser. Of course, the moniker “very narrow” must somehow be related to the application. For spectroscopic systems (e.g., RF or DIAL), a requisite spectral width might be 80ı tilted structures. Opt. Lett. 31, 1193–1195 (2006)

CO2 -Laser-Inscribed Long Period Fiber Gratings: From Fabrication to Applications

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Yiping Wang and Jun He

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CO2 Laser Inscription Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LPFG Inscription in Conventional Glass Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LPFG Inscription in Solid-Core PCFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LPFG Inscription in Air-Core PBFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grating Formation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Refractive Index Modulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymmetrical Mode Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Improvements of Grating Fabrications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pretreatment Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Posttreatment Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensing Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strain Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bend Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Torsion Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biochemical Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Communication Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Band-Rejection Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gain Equalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polarizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Y. Wang () · J. He Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province, College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen, China Guangdong and Hong Kong Joint Research Centre for Optical Fibre Sensors, Shenzhen University, Shenzhen, China e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2019 G.-D. Peng (ed.), Handbook of Optical Fibers, https://doi.org/10.1007/978-981-10-7087-7_78

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Couplers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mode Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter presents a systematic review of long period fiber gratings (LPFGs) inscribed by CO2 laser irradiation. Firstly, various fabrication techniques based on CO2 laser irradiations are introduced that inscribe LPFGs in different types of optical fibers such as conventional glass fibers, solid-core photonic crystal fibers, and air-core photonic bandgap fibers. Secondly, possible mechanisms, including residual stress relaxation, glass structure changes, and physical deformation, of refractive index modulations in the CO2 -laser-induced LPFGs are discussed. Asymmetrical mode coupling, resulting from single-side laser irradiation, is analyzed to understand the unique optical properties of the CO2 -laser-induced LPFGs. Thirdly, several pre- and posttreatment techniques for enhancing the grating efficiency are described. Fourthly, sensing applications of CO2 -laserinduced LPFGs for temperature, strain, bend, torsion, pressure, and biochemical sensors are presented. Finally, communication applications of CO2 -laser-induced LPFGs, such as band-rejection filters, gain equalizers, polarizers, couplers, and mode converters, are presented and discussed. Keywords

Gratings · Long period fiber gratings (LPFGs) · Optical fiber sensors · CO2 laser

Introduction Optical fiber gratings play a vital role in the field of optical communications and sensors. There are two types of in-fiber gratings: fiber Bragg gratings (FBGs) with period or pitch in the order of optical wavelength (Hill et al. 1978; Rao 1997, 1999; Kersey et al. 1997), and long period fiber gratings (LPFGs) with period or pitch in hundreds of wavelengths (Vengsarkar et al. 1996a, b; Bhatia and Vengsarkar 1996; Erdogan 1997; Eggleton et al. 1997, 1999, 2000; James and Tatam 2003; Bhatia 1999; Davis et al. 1998a, b). Since Hill et al. (1978) and Vengsarkar et al. (1996a) wrote the first FBG and LPFG in conventional glass fibers in 1978 and 1996, respectively, the fabrication and application of in-fiber gratings have developed rapidly. Various fabrication methods, such as ultraviolet (UV) laser exposure (Vengsarkar et al. 1996a, b; Bhatia and Vengsarkar 1996; Erdogan 1997; Eggleton et al. 1997, 1999, 2000; James and Tatam 2003; Bhatia 1999), CO2 laser irradiation (Davis et al. 1998a, b; 1999; Kakarantzas et al. 2001, 2002; Rao et al. 2003a, 2004; Liu and Chiang 2008a; Wang et al. 2006a, b, 2007a, 2008a; Zhong

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et al. 2014a, b, 2015; Fu et al. 2015; Tan et al. 2013; Geng et al. 2012; Yang et al. 2011, 2017; Tian et al. 2013; Zhu et al. 2003a, 2004; Wu et al. 2011; Tang et al. 2015, 2017), electric arc discharge (Kosinski and Vengsarkar 1998; Karpov et al. 1998; Hwang et al. 1999; Humbert et al. 2003; Dobb et al. 2004; Rego et al. 2006), femtosecond laser exposure (Kondo et al. 1999; Fertein et al. 2001), mechanical microbends (Savin et al. 2000; S et al. 2001; Steinvurzel et al. 2006a; Su et al. 2005), etched corrugations (Lin and Wang 1999, 2001; Lin et al. 2001a, b, 2002; Yan et al. 2002), and ion beam implantation (Fujimaki et al. 2000; Bibra et al. 2001; Bibra and Roberts 2000), have been demonstrated to write LPFGs in different types of optical fibers. Numerous LPFG-based devices have also been developed for sensing and communication applications (Kersey et al. 1997; Davis et al. 1998a; Rao et al. 2004). Davis et al. reported gratings written by CO2 laser irradiation in a conventional glass fiber in 1998 (Davis et al. 1998a, b). Compared with the UV laser exposure, CO2 laser irradiation is more flexible and is of lower cost because no photosensitivity and any other pretreated processes are required to write a grating in the glass fibers (Davis et al. 1998a, b, 1999; Kakarantzas et al. 2001, 2002; Rao et al. 2003a, 2004; Liu and Chiang 2008a; Wang et al. 2006a, b, 2007a, 2008a; Zhong et al. 2014a, b, 2015; Fu et al. 2015; Tan et al. 2013; Geng et al. 2012; Yang et al. 2011, 2017; Tian et al. 2013; Zhu et al. 2003a, 2004; Wu et al. 2011; Tang et al. 2015, 2017). Moreover, the CO2 laser irradiation process can be controlled to generate complicated grating profiles via point-to-point exposure without any expensive masks. Hence, this technique could be used to write LPFGs in almost all types of fibers including pure-silica photonic crystal fibers (PCFs). Liu et al. presented development on the CO2 -laser-induced LPFGs (Liu and Chiang 2008a). The most exciting development of writing with CO2 laser irradiation is to successfully inscribe the LPFG in an air-core photonic bandgap fiber (PBF) by using focused CO2 laser light to periodically perturb or deform the air holes along the fiber axis (Wang et al. 2008a). Here we present a systematic review of LPFGs written with CO2 laser irradiation in different types of optical fibers. The methods for writing LPFGs in conventional glass fibers, solid-core PCFs, and air-core PBFs with a CO2 laser are presented in section “CO2 Laser Inscription Techniques.” Then possible grating formation mechanisms, for example, residual stress relaxation, glass structural changes, and physical deformation that give rise to the refractive index modulations of the CO2 laser-induced LPFGs, are analyzed in section “Grating Formation Mechanisms.” Asymmetrical mode coupling in LPFGs is also discussed in this section. Various methods for fabricating better LPFGs, including pretreatment methods such as hydrogen loading and applying prestrain and posttreatment methods such as applying tensile strain and changing temperature, are described in section “Improvements of Grating Fabrications.” Subsequently, sensing applications such as temperature, strain, bend, torsion, pressure, and biochemical sensors and communication applications such as band-rejection filters, gain equalizers, polarizers, couplers, and mode converters using the CO2 -laser-induced LPFGs are presented in sections “Sensing Applications” and “Communication Applications,” respectively.

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CO2 Laser Inscription Techniques Since Davis et al. reported the CO2 -laser-induced LPFG in a conventional glass fiber in 1998 (Davis et al. 1998a, b), various CO2 laser irradiation techniques have been demonstrated and improved to write high-quality LPFGs in different types of optical fibers, including conventional glass fibers, PCFs, and PBFs, and to achieve unique grating properties. This section reviews the use of CO2 laser irradiation for writing LPFGs in conventional glass fibers, solid-core PCFs, and air-core PBFs.

LPFG Inscription in Conventional Glass Fibers Typically, in most of LPFG fabrication setups employing a CO2 laser (Davis et al. 1998a, b; Zhu et al. 2003a; Braiwish et al. 2004), as shown in Fig. 1, the fiber is periodically moved along its axis direction via a computer-controlled translation stage, and the CO2 laser beam irradiates periodically the fiber through a shutter controlled by the same computer. A light source and an optical spectrum analyzer are employed to monitor the evolution of the grating spectrum during the laser irradiation. This is a typical point-to-point technique for writing a grating in an optical fiber. Such an LPFG fabrication system usually requires an exact controlling of both the shutter and the translation stage to achieve a good simultaneousness of the laser irradiation and the fiber movement. Moreover, vibration from the periodic movement of the fiber during the irradiation with the CO2 laser beam could affect the stability and repeatability of grating fabrication. Rao et al. demonstrated a novel grating fabrication system based on twodimensional scanning of the CO2 laser beam (Rao et al. 2003a, 2004), as shown in Fig. 2. One end of the fiber is fixed and the other is attached to a small weight to provide a constant strain in the fiber. The strain could make it easier to fabricate LPFGs and improve the efficiency of the grating fabrication, as discussed in section “Sensing Applications” The focused high-frequency CO2 laser pulses scanned periodically across the employed fiber along “x” direction and then shifted a grating pitch along “y” direction (the fiber axis) to create next grating period by means of two-dimensional optical scanners under the computer’s control. Compared with typical point-to-point fabrication systems (Davis et al. 1998a, b; Zhu et al. 2003a; Braiwish et al. 2004), no synchronization is required in such a system because the Fig. 1 Schematic diagram of an LPFG fabrication based on the point-to-point technique employing a CO2 laser

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Fig. 2 Schematic diagram of LPFG fabrication based on two-dimensional scanning of focused high-frequency CO2 laser pulses (Rao et al. 2003a)

Fig. 3 Schematic diagram of an improved LPFG fabrication based on the point-to-point technique employing a CO2 laser

employed fiber is not periodically moved along the fiber axis. Such a system can write high-quality LPFGs with nearly zero insertion loss. A similar fabrication setup was also demonstrated to write an LPFG with a CO2 laser, in which the laser beam, however, scanned one-dimensionally across the fiber that was periodically moved along the fiber axis (Chan et al. 2007). The authors developed an improved LPFG fabrication system based on the pointto-point technique employing a CO2 laser, as shown in Fig. 3, combining the advantages of the two fabrication systems illustrated in Figs. 1 and 2. The CO2 laser beam is, through a shutter and a mirror, focused on the fiber by a cylindrical lens with a focus length of 254 mm. Both the mirror and the lens are mounted on a linear air-bearing motor stage (ABL 1500 from Aerotech). A LabVIEW program has been developed to control simultaneously the operations of both the linear motor stage and the shutter so that the fiber is exposed once as soon as the focused laser beam is shifted by a grating pitch via the mirror. In other words, the fiber is not moved in this system, which overcomes the disadvantage of the fiber vibration, resulting from periodic movement of the fiber, in normal point-to-point grating fabrication setups shown in Fig. 1. Zhong et al. also developed a promising LPFG inscribing system based on an improved two-dimensional scanning technique with a focused CO2 laser beam (Zhong et al. 2014a), as shown in Fig. 4a. This system consists of an industrial CO2 laser with a maximum power of 10 W (SYNRAD 48-1) and a power stability

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Fig. 4 (a) Schematic diagram of an improved LPFG fabrication system based on a twodimensional scanning technique using a CO2 laser. (b) Operation interface of the LPFG fabrication system (Zhong et al. 2014a)

of ˙10%, an electric shutter for turning on/off the laser beam, an infrared ZNSE PO/CX lens with a focused length of 63.5 mm, a four-times beam expander for decreasing the diameter of the focused laser spot, and a three-dimensional ultraprecision motorized stage (Newport XMS50,VP-25X, and GTS30V) with a minimum incremental motion of 10 nm and a bi-directional repeatability of 80 nm. A closed loop control system is employed to improve the power stability of the CO2 laser to ˙2%, which is a huge advantage of this LPFG inscribing system. The power stability (˙2%) of the CO2 laser improves effectively the stability and reproducibility of grating inscription. For example, the success rate of grating inscription is almost 100% in this system. In contrast, the success rate was about 30% in the system with a CO2 laser with power stability of ˙10% (Rao et al. 2003a, 2004). A supercontinuum light source (NKT Photonics SuperK™ Compact) and an optical spectrum analyzer (YOKOGAWA AQ6370C) are employed to monitor the transmission spectrum of the CO2 -laser-inscribed LPFG during grating inscription. A control program with an easy-to-use operation interface is developed by use of LabVIEW software in order to control every device in the system and to inscribe

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Fig. 5 Photograph of an asymmetric LPFG with periodic grooves (Wang et al. 2006a)

high-quality LPFGs. As soon as the grating parameters, such as grating pitch, number of grating periods, number of scanning cycles, are entered via the operation interface illustrated in Fig. 4b and the “Write” button is clicked, a high-quality LPFG could be obtained. CO2 laser irradiation may cause unexpected physical deformation, resulting from laser heating, of fiber structures during LPFG fabrication. Such physical deformations are usually avoided to decrease the insertion loss of the written LPFGs during early grating fabrications with a CO2 laser (Davis et al. 1998a, b; Rao et al. 2003a). Wang et al. reported a novel technique for writing an asymmetric LPFG by means of carving periodic grooves on the surface of an optical fiber with a focused CO2 laser beam (Wang et al. 2006a), as shown in Fig. 5. Physical deformations, i.e., periodic grooves, in such an asymmetric LPFG, do not cause a large insertion loss because these grooves are totally confined within the outer cladding and have no influence on the light transmission in the fiber core. Moreover, such grooves enhance the efficiency of grating fabrication and introduce unique optical properties, e.g., extremely high strain sensitivity, into the gratings (Wang et al. 2006a; 2007b). Further investigations discover that the insertion loss of LPFGs is mainly due to the nonperiodicity and the disorder of refractive index modulations in the gratings. Asymmetric LPFGs were also fabricated in thin core fibers using the same method (Fu et al. 2015). The proposed thin-core LPFG exhibits a high extinction ratio of over 25 dB at the resonant wavelength and a narrowed 3 dB-bandwidth of 8.7 nm, which is nearly one order of magnitude smaller than that of LPFGs in conventional single mode fibers. It also exhibits high polarization-dependent loss of over 20 dB at resonant wavelength. Additionally, a few other CO2 laser irradiation methods have been demonstrated to write specialty LPFGs. For example, (1) edge-writing was reported to inscribe LPFGs with high-frequency CO2 laser pulses (Zhu et al. 2007a, b, 2009a). Refractive index disturbance in such an edge-written LPFG mainly occurs at the edge region of the fiber cladding rather than in the fiber core. (2) A helical LPFG was fabricated in an optical fiber that continuously rotates and moves along the fiber axis during CO2 laser irradiation (Oh et al. 2004a). Compared with a conventional LPFG, a helical LPFG has a very low polarization dependent loss (PDL) due to a screwtype index modulation in this grating, which thus could be a potential technique for achieving an LPFG with low PDL. (3) Microtaper-based LPFGs were also fabricated by means of tapering periodically a conventional glass fiber with a focused

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CO2 laser beam (Kakarantzas et al. 2001; Zhong et al. 2014b; Xuan et al. 2009; Zhu et al. 2005a; Lei et al. 2006). Negative temperature coefficient of the resonant wavelength was observed in the microtaper-based LPFGs (Xuan et al. 2009). CO2 laser irradiation has also been used to fabricate microtaper-based LPFGs on soft glass fibers. Soft glass fibers are developed to achieve flattened and near-zero dispersion profile and have potential bend sensing applications due to their unique light transmission ability. An all-solid soft glass fiber consisting of two types of commercial lead silicate glasses (core: Schott SF57 and cladding: Schott SF6) was designed and drawn. As shown in Fig. 6a, the core and cladding diameters of this fiber are 2.4 and 175 m, respectively. Refractive index in the core and cladding are 1.80 and 1.76, respectively. The measured transmission loss of the soft glass fiber is about 6.1 dB per meter at the wavelength of 1550 nm. Compared with conventional silica glass fibers, the soft glass fiber has a lower drawing temperature of about 600–700 ı C. The soft glass fiber was tapered periodically the by use of a CO2 laser grating fabrication system, as shown in Fig. 6b. Such a soft glass fiber with periodic microtapers could be used to develop promising bend sensors with a sensitivity of 27.75 W/m1 by means of measuring the bend-induced change of light intensity. The proposed bend sensor exhibits a very low measurement error of down to ˙1% (Wang et al. 2012). Apart from conventional single mode fibers (SMFs), there have been a number of studies on CO2 -laser writing of LPFGs in boron-doped fibers (Grubsky and Feinberg 2005; Liu et al. 2009a; Kim et al. 2001, 2002). The grating writing efficiency can be enhanced with a fiber annealed at a sufficiently high temperature (Liu and Chiang 2008a; Grubsky and Feinberg 2005; Liu et al. 2009a). Compared with normal UV laser exposure technique, the CO2 laser irradiation technique is easily used to write special gratings such as phase-shifted LPFGs (Zhu et al. 2004, 2005a), chirped LPFGs (Yan et al. 2008), complicated apodized LPFGs (Gu et al. 2009), grating pairs (Chan et al. 2007), and ultra-long period fiber gratings with a period of up to several millimeters (Zhu et al. 2007c, 2005b; Rao et al. 2006). LPFGs have also been successfully achieved with a CO2 laser during the fiber-drawing process (Hirose et al. 2007). So, it is possible for the CO2 laser irradiation technique to continuously write numerous LPFGs with high quality and low cost in an optical fiber to develop potential distributed sensing systems and communication applications. By the way,

Fig. 6 (a) Cross-section microscope image of the all-solid soft glass fiber. (b) Microscope image of the soft glass fiber with periodic microtapers created by CO2 laser irradiation technique (Wang et al. 2012)

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CO2 laser irradiation can be used to enhance the UV photosensitivity of GeO2 :SiO2 optical fibers before grating writing (Brambilla et al. 1999).

LPFG Inscription in Solid-Core PCFs Over the past decade, PCFs have attracted a great deal of interest due to their unique microstructures and optical properties (Knight et al. 1996; Smith et al. 2003). Since Eggleton et al. reported the grating in a photosensitive PCF with a Ge-doped core in 1999 (James and Tatam 2003), a large number of gratings have been written in different types of PCFs with or without photosensitivity by the use of various fabrication techniques such as UV laser exposure (Bhatia 1999), CO2 laser irradiation (Kakarantzas et al. 2002; Wang et al. 2006b, 2007a; Tian et al. 2013; Zhong et al. 2014b, 2015; Zhu et al. 2003a, 2004; Yang et al. 2017), electricarc discharge (Zhu et al. 2004), femtosecond laser exposure (Karpov et al. 1998; Humbert et al. 2003), and two-photon absorption (Groothoff et al. 2003). UV laser exposure is a common technique for writing a FBG/LPFG in a Ge-doped PCF with a photosensitivity (James and Tatam 2003; Wang et al. 2009a, b). In contrast, CO2 laser irradiation is a highly efficient, low cost technique for writing an LPFG in a pure-silica PCF without photosensitivity (Kakarantzas et al. 2002; Wang et al. 2006b, 2007a; Tian et al. 2013; Zhong et al. 2014b, 2015; Zhu et al. 2003a, 2004; Yang et al. 2017). Kakarantzas et al. reported, as shown in Fig. 7, the example of structural LPFGs written in pure-silica solid-core PCFs (Kakarantzas et al. 2002, 2003). The gratings are realized by periodic collapse of air holes in the PCF via heat treatment with a CO2 laser. The resulting periodic hole-size perturbation produces core-to-claddingmode conversion, thus creating a novel LPFG in the PCF (Kakarantzas et al. 2002). This technique, combining with periodic mechanical twisting, can be used to fabricate a rocking filter in a polarization-maintaining PCF (Kakarantzas et al. 2003). As shown in Fig. 8, an asymmetrical LPFG with periodic grooves was written in a pure-silica large-mode-area PCF by the use of a focused CO2 laser beam (Wang et al. 2006b, 2007a). The repeated scanning of the focused CO2 laser beam creates a local high temperature in the fiber, which leads to the collapse of air holes

Fig. 7 LPFG written in a pure-silica solid-core PCF with a CO2 laser (Kakarantzas et al. 2002)

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Fig. 8 Asymmetrical LPFG with periodic grooves in a pure-silica PCF (Wang et al. 2007a)

and the gasification of SiO2 on the fiber surface. Consequently, periodic grooves with a depth of about 15 m and a width of about 50 m are created on the fiber, as shown in Fig. 8b. Such grooves, especially collapse of air holes, induce periodic refractive index modulations along the fiber axis due to the well-known photoelastic effect, thus creating an LPFG in the PCF. This asymmetrical LPFG has unique optical properties, e.g., high strain sensitivity, low temperature sensitivity, and high polarization dependence, as discussed in sections “Sensing Applications” and “Communication Applications.” As shown in Fig. 9, an inflated LPFG (I-LPFG) was also inscribed in a pure-silica PCF and could be used as a high-sensitivity strain sensor and a high-sensitivity gas pressure sensor (Zhong et al. 2014b, 2015). The I-LPFG was inscribed by use of the pressure-assisted CO2 laser beam scanning technique to inflate periodically air holes of a PCF. Such periodic inflations enhanced the sensitivity of the LPFG-based strain sensor to 5.62 pm/". After high temperature annealing of the I-LPFG, moreover, a good repeatability and stability of temperature response with a sensitivity of 11.92 pm/ı C was achieved (Zhong et al. 2014b). In addition, the I-LPFG with periodic inflations exhibits a very high gas pressure sensitivity of 1.68 nm/MPa, which is one order of magnitude higher than that, i.e., 0.12 nm/MPa, of the LPFG without periodic inflations. Moreover, the I-LPFG has a very low temperature sensitivity of 2.83.1 pm/ı C due to the pure silica material in the PCF so that the pressure measurement error, resulting from the cross sensitivity between temperature and gas pressure, is less than 1.71.8 kPa/ı C without temperature compensation. So, the I-LPFG could be used to develop a promising gas pressure sensor, and the achieved pressure measurement range is up to 10 MPa (Zhong et al. 2015).

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Fig. 9 Microscope image of the cross section of the PCF (a) before and (b) after CO2 laser irradiation, (c) side view of the CO2 -laser-inscribed inflated LPFG with periodic inflations (Zhong et al. 2015)

In addition to the asymmetrical LPFGs inscribed in PCFs, the symmetrical LPFGs were also inscribed in PCFs using the same method (Tian et al. 2013). Results show that symmetric index perturbation induced by laser irradiation with the aid of a 120ı gold-coated reflecting mirror results in LP0n symmetric mode coupling, while asymmetric irradiation without using the mirror leads to LP1n asymmetric mode coupling. Because of the azimuthally anisotropic hexagonal cladding structure, symmetric irradiation yields far more reproducible LPFGs in PCFs than asymmetric irradiation. On the other hand, the irradiation symmetry has little effect on the reproducibility of LPFGs inscribed in SMFs due to the isotropy of its all-solid cladding structure. Furthermore, a highly compact LPFG with only 8 periods and a short total length of 2.8 mm was written in a pure-silica large-mode-area PCF by the common point-by-point technique employing a CO2 laser (Zhu et al. 2003a), in which clear physical deformation was also observed. In contrast, another LPFG without geometrical deformation and fiber elongation was written in an endlessly singlemode PCF by periodic stress relaxation resulting from CO2 laser irradiation (Zhu et al. 2005c). Moreover, an LPFG pair has been successfully created in a pure-silica PCF with a CO2 laser to develop a stain sensor (Shin et al. 2009). A novel coupled local-mode theory could be used to model and analyze this type of PCF-based LPFGs with periodic collapses of air holes (Jin et al. 2010). Such a theory is based on calculating the variations of local-mode profiles and propagation constants over the perturbed regions and on solving the coupled local-mode equations to obtain a quantitative description of the intermodal energy exchange. The CO2 laser irradiation technique could also be used to write an LPFG in a conventional or photonic crystal polarization-maintaining (PM) fibers (Lee et al. 2008). Such an LPFG has two clean polarization-splitting rejection bands. The writing efficiencies of gratings in the two types of PM fibers depend strongly on the fiber orientation, with the highest efficiency obtained when the irradiation direction is along the slow axis of the fiber (Lee et al. 2008). Such orientation dependence is much stronger for a conventional PM fiber than for a photonic crystal PM fiber

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and is attributed to the relaxation of mechanical stress in the stress-applying parts of the fiber.

LPFG Inscription in Air-Core PBFs As discussed above, a large number of gratings have been demonstrated in different types of PCFs by the use of various fabrication techniques (Zhu et al. 2003a, 2004; Karpov et al. 1998; Humbert et al. 2003; Groothoff et al. 2003). All of these gratings, however, were written in index-guiding PCFs, instead of bandgap-guiding fibers. Recently, PBF-based gratings were also written in a new kind of bandgap-guiding fibers such as fluid-filled PBFs (Steinvurzel et al. 2006a, b; Wang et al. 2009c; Iredale et al. 2006; Kuhlmey et al. 2009) and all-solid PBFs (Jin et al. 2007a, b). However, PBF-based gratings have not been reported in air-core PBFs until the success in writing a high-quality LPFG in an air-core PBF (Wang et al. 2008a). After that, these air-core PBF-based LPFGs were further developed as gas-pressure sensors by the use of gas-pressure-induced deformation on the LPFG (Tang et al. 2015) or refractive index change in the air core of the PBF (Tang et al. 2017). Since almost 100% of the light propagates in the air holes of an air-core PBF and not in the glass (Smith et al. 2003), PBF-based gratings offer a number of unique features including: high dispersion, low nonlinearity, reduced environmental sensitivity, unusual mode coupling, and new possibilities for long-distance light-matter interactions (by incorporating additional materials into the air-holes). Bandgap-based grating in air-core PBFs, therefore, represent an important platform technology with manifest applications in areas such as communications, fiber lasers and sensing. Periodic index modulations are usually required to realize mode coupling in in-fiber gratings. Although this presents no difficulties in conventional glass fibers (Davis et al. 1998a; Rao et al. 2003a), solid-core PCFs (James and Tatam 2003; Kakarantzas et al. 2002; Wang et al. 2007a), and solid-core PBFs (Steinvurzel et al. 2006a, b; Iredale et al. 2006; Kuhlmey et al. 2009; Jin et al. 2007a, b), it is very difficult, even impossible, to directly induce index modulations in an air-core PBF due to the air core structure, thereby seriously obstructing the development of PBF-based gratings over the past decade. Wang et al. reported gratings written in an air-core PBF by the use of a focused CO2 laser beam to periodically deform/perturb air holes along the fiber axis in 2008 (Wang et al. 2008a), as shown in Fig. 10. This reveals that it is experimentally possible to write a grating in an air-core PBF. Both the excellent stability of CO2 laser power and the good repeatability of optical scanning are very critical to writing a high-quality grating in an air-core PBF. An experimental setup being similar to that in Fig. 2 was used to write an LPFG in an air-core PBF (Crystal-Fiber’s HC-1500-02). Compared with the fabrication parameters for writing a grating in a solid-core PCF (Wang et al. 2006b, 2007a), a lower average laser power of about 0.2 W and shorter total time of laser irradiation are used to write an LPFG in an air-core PBF (Wang et al. 2008a). The focused CO2 beam scans periodically the PBF with a line speed of scanning of 2.9 mm/s, causing the ablation of glass on the fiber surface and the partial or complete collapse of air holes in the cladding due

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Fig. 10 Cross-section image of an air-core PBF (a) before and (b) after CO2 laser irradiating, (c) side image of an LPFG written in the air-core PBF, where about two periods of the LPFG are illustrated (Wang et al. 2008a)

to the CO2 -laser-induced local high temperature, as shown in Fig. 10. The outer rings of air holes in the cladding, facing to the CO2 laser irradiation, were largely deformed; however, little or no deformation were observed in the innermost ring of air-holes and in the air core. As a result, periodic index modulations are achieved along the fiber axis due to the photoelastic effect, thus creating a novel LPFG in the air-core PBF. For the LPFG written in air-core PBF, periodic perturbations of the waveguide (geometric) structure could be the dominant factor that causes resonant mode coupling, although the stress-relaxation-induced index variation may also contribute a little. Wu et al. reported the inscription of high-quality LPFGs in simplified hollowcore PBFs using CO2 -laser-irradiation method (Wu et al. 2011). The PBFs are composed of a hollow hexagonal core and six crown-like air holes. These LPFGs are originated from the strong mode-coupling between the LP01 and LP11 core modes. A dominant physical mechanism for the mode coupling is confirmed to be the periodic microbends rather than the deformations of the cross section or other common factors. In addition, the LPFGs are highly sensitive to strain and nearly insensitive to temperature and are promising candidates for gas sensors and nonlinear optical devices. Normal LPFGs written in the index-guiding fibers have a positive relationship between resonant wavelength and grating pitch (Vengsarkar et al. 1996a, b; Erdogan 1997; Eggleton et al. 1997). In contrast, the LPFGs written in the bandgap-guiding

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Fig. 11 (a) Transmission spectra of six LPFGs, with different grating pitches, written in an air-core PBF. (b) The relationship between the pitch of each LPFG and the corresponding research wavelength, where two attenuation dips for each LPFG are observed from 1500 to 1680 nm, indicating that the fundamental mode is coupled to two different higher order modes (Wang et al. 2008a)

air-core PBF have distinct phase matching condition as function of wavelength. As shown in Fig. 11, the resonant wavelengths of the LPFGs written in an air-core PBF decrease with the increase in grating pitch, which is opposite to the LPFGs written in the index-guiding fibers (Vengsarkar et al. 1996a, b). Moreover, this PBF-based LPFG has unique optical properties such as very large PDL, large strain sensitivity, and very small sensitivity or insensitivity to temperature, bend and external refractive index, as shown in Fig. 4 in Wang et al. (2008a). Further investigations are being done to well understand resonant mode coupling and unique optical properties in the gratings written in air-core PBFs.

Grating Formation Mechanisms Refractive Index Modulations Possible mechanisms for refractive index modulation in the CO2 -laser-induced LPFGs could be attributed to residual stress relaxation (Kim et al. 2001, 2002; Li et al. 2008), glass densification (Grubsky and Feinberg 2005; Liu et al. 2009a; Morishita and Kaino 2005; Morishita et al. 2002), and/or physical deformation, depending on the types of the employed fibers and on the practical fabrication techniques. A few methods have been demonstrated to measure the refractive index modulation in the CO2 -laser-induced LPFGs (Li et al. 2008; Hirose et al. 2008).

Residual Stress Relaxation Residual stress relaxation is found to be the main mechanism for the refractiveindex change in the CO2 -laser-induced LPFGs written in optical fibers drawn at high drawing forces (Liu et al. 2009a; Kim et al. 2001, 2002; Li et al. 2008). Residual stress is formed in optical fibers during the drawing process, resulting

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mainly from a superposition of thermal stress caused by a difference in thermal expansion coefficients between core and cladding and mechanical stress caused by a difference in the viscoelastic properties of the two regions (Kim et al. 2001, 2002; Paek and Kurkjian 1975). Such residual stress can change refractive index in the fibers through the stress-optic effect and thus affect the optical properties of the fibers. Mechanical stress in a CO2 -laser-induced LPFG written in a Ge-Bcodoped fiber can be fully relaxed by CO2 laser irradiation. Consequently, only thermal stress remains in the fiber core due to a mismatch of the thermal expansion coefficients of the fiber core and cladding. Residual stress relaxation usually results in a decrease of refractive index in the fibers. And the efficiency of refractive-index decrease depends strongly on the types of fiber and can be enhanced linearly with the drawing force during the drawing process of the fiber (Kim et al. 2001, 2002). For example, the refractive-index change, resulting from residual stress relaxation, in a Ge-B-codoped fiber drawn at 0.53, 1.38, 2.50, and 3.43 N was measured to be 3.6  105 , 8.0  105 , 1.7  104 , and  2.1  104 , respectively (Kim et al. 2002). Another experiment shows that residual stress relaxation results in a larger refractive index change of 7.2  104 in a Corning SMF-28e fiber (Li et al. 2008).

Glass Structure Change The changes of glass structure (glass volume increase and glass densification) play the dominant mechanisms in the CO2 -laser-induced LPFGs written in a commercial boron-doped single-mode fiber (Grubsky and Feinberg 2005; Liu et al. 2009a; Morishita and Kaino 2005; Morishita et al. 2002). For an unannealed fiber or a fiber annealed at a temperature lower than about 380 ı C, glass densification, resulting in an increase in the refractive index, plays the dominant role (Liu et al. 2009a). For a fiber annealed at a higher temperature than about 400 ı C, however, glass volume increase, resulting in a decrease in the refractive index, becomes more important (Liu et al. 2009a). On the other hand, residual stress relaxation in the fiber core, which is the dominant mechanism in a conventional Ge-doped or Ge-B-codoped fiber, plays only a minor role in the boron-doped fiber that has a core with a small residual stress and a low fictive temperature. For the CO2 -laser-induced LPFGs in the Ge-doped or Ge-B-codoped fibers, e.g., Corning SMF-28 fiber, with large residual stress, the resonance wavelength shifts toward the shorter wavelength with the increase of the laser exposure dose due to the negative index modulation resulting from residual stress relaxation (Kim et al. 2001, 2002; Li et al. 2008). For the CO2 -laser-induced LPFGs in the boron-doped fibers with small residual stress, on the contrary, the resonance wavelength shifts toward the longer wavelength with the increase of the laser exposure dose due to the positive index modulation resulting from glass densification (Grubsky and Feinberg 2005; Liu et al. 2009a; Morishita and Kaino 2005; Morishita et al. 2002). Physical Deformation Physical deformation is considered as one of the main mechanisms of CO2 -laserinduced LPFGs. During CO2 laser irradiation, the fiber usually elongates or tapers based on the so-called “self-regulating” mechanism (Grellier et al. 1998), resulting

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from constant axial tension and the CO2 -laser-induced local high temperature in the fiber. Thus, the fiber diameter decreases and eventually reaches a critical point at which the fiber elongation stops because no sufficient energy is absorbed to keep the softening temperature. Such physical deformation induces a change of effective refractive index in the fiber, and thus LPFGs are created in the periodically taped optical fibers (Kakarantzas et al. 2001; Xuan et al. 2009; Zhu et al. 2005a; Lei et al. 2006). Moreover, the CO2 -laser-induced high temperature in the fiber causes, not only the fiber elongation and the diameter decrease but also the ablation of glass on the fiber surface and the partial or complete collapses of air holes in the PCFs. As a result, different types of LPFGs are written in conventional glass fibers in which periodic grooves/microbends are created on the fiber surface (Kakarantzas et al. 2001; Wang et al. 2006a; Xuan et al. 2009; Zhu et al. 2005a; Lei et al. 2006), and in microstructured optical fibers, such as solid-core PCFs (Kakarantzas et al. 2002; Wang et al. 2006b, 2007a; Zhu et al. 2003a) and air-core PBF (Wang et al. 2008a), in which air holes are partially or completely collapsed. Theoretical and experimental investigations show that the micro-deformationinduced LPFGs in pure-silica PCFs have discrete attenuation peaks whose spectral positions are correlated to the beat length between the fundamental mode and the first higher-order mode (Nielsen et al. 2003). Thus, simple description of the modal properties based on the perfectly uniform fiber structure may explain the modecoupling properties of the micro-deformation-induced LPFGs in pure-silica PCFs (Nielsen et al. 2003).

Asymmetrical Mode Coupling Asymmetrical mode coupling is one of the most distinctive features in the CO2 laser-induced LPFGs, resulting in unique optical properties that have a large number of promising sensing and communication applications (Rao et al. 2003a; Wang et al. 2006a), as discussed in sections “Sensing Applications” and “Communication Applications.” As shown in Fig. 12a, the laser energy is strongly absorbed on the incident side of the fiber, while the CO2 laser light (œ D 10.6 m) illuminates on the optical fiber (VanWiggeren et al. 2000). The nonuniform absorption results in an asymmetrical refractive index profile within the cross section of the CO2 laser-induced LPFGs. In other words, a larger refractive index change is induced in the incident side of the fiber than in the opposite side. Consequently, as shown in Fig. 12b, asymmetrical mode coupling occurs in the CO2 -laser-induced LPFGs (Davis et al. 1998b; Wang et al. 2008b; Slavík 2007; Ryu et al. 2003a, b) so that a higher PDL usually are observed in the gratings (Wang et al. 2007a, b; Ryu et al. 2003a). Such asymmetrical mode coupling also results that the responses of the CO2 -laser-induced LPFGs to the applied bending (Rao et al. 2003a; VanWiggeren et al. 2000, 2001; Wang et al. 2004; Wang and Rao 2005), twisting (Wang and Rao 2004a, b; Wang et al. 2005), and transverse loading (Wang et al. 2007c, 2003a; Rao et al. 2003b) depend strongly on the fiber orientations, which is distinct from the gratings written by the UV exposure technique (Bhatia and Vengsarkar 1996; James and Tatam 2003; Bhatia 1999; Davis et al. 1998a). Moreover, the asymmetry in the

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Fig. 12 (a) Calculated intensities (relative to unity incident intensity) within the cross section of optical fiber during the single-side irradiation of CO2 laser light (VanWiggeren et al. 2000). (b) Asymmetrical mode coupling at the resonant wavelength in the CO2 -laser-induced LPFG (Wang et al. 2008b)

stress distribution of the cladding is found to be much larger than that in the core of the CO2 -laser-induced LPFG (Ryu et al. 2003a). So, the polarization-dependent transmission characteristics of the LPFG are affected mostly by the asymmetric stress distribution in the cladding region rather than the core region. To solve the problem of asymmetrical mode coupling in LPFGs written by onesided exposure of a CO2 laser beam, various methods based on symmetric exposure of the laser beam were demonstrated to achieve axially symmetric LPFG by rotating the fiber during exposure (Oh et al. 2004b), by focusing a ring-shaped CO2 laser beam axially on the fiber with a concave mirror (Oh et al. 2004b), and by placing a special reflector behind the fiber (Grubsky and Feinberg 2006). The achieved LPFGs with an axially symmetric refractive index profile exhibit clean spectra, very low insertion loss (neff >nou and as a core mode for neff >ncl . Modes corresponding to the solutions in Eq. and the appropriate boundary  9 dE z dH z conditions [the continuity of Ez (Hz ) and d  d  ] can be classified as either TE modes (Ez D 0), TM modes (Hz D 0), or hybrid modes (Ez ¤ 0, Hz ¤ 0), respectively. The solutions of Eq. 9 in each region of the fiber depend on the magnitude of neff with respect to the refractive index n. If neff n, the two independent solutions are the vth -order-modified Bessel functions of the first and second kinds Iv , and Kv , respectively. Love has presented some solutions in ref. (Love et al. 1991). When the diameter of the MNOF is very small or the cladding is very large (rco 150 m is finally demonstrated. As mentioned above, the proposal of using femtosecond laser as the optical source may be a good choice for longer measurement range if combining the noise-like phase modulation method. With the noise-like phase modulation by an arbitrary waveform generator, the signals from the detection arm and the reference arm which have been synchronized are added and yield a correlation gating at an arbitrary delay. By carefully adjusting the gating delay, a measurement range beyond 1 km can be reached. Nevertheless, as the spatial resolution and measurement range increases, the effects of chromatic dispersion should be taken into consideration. For example, when the spatial resolution is about 10 m, the measurement range cannot exceed several meters using the phase modulation method mentioned above. However, the method of using additional delay fiber may cancel the chromatic dispersion effects. Reflection Sensitivity and Phase Noise The measurement reflection sensitivity of an OLCR system is decided by optical source and the receiver. The source power finally determines the amount of the power reflected in the detection and reference arm and the bandwidth of the receiver determines the amount of the noise. The SNR can be expressed as (Sorin and Baney 1992): SNR D

4kT f =Reff

2 64: D p !  2 2 L

(63)

Note that, since the S-matrix is a square matrix, L different sequences of L bits each must be used. This, however, does not represent a penalty to the measurement time of the system, since the coding gain in Eq. 63 already compares the SNR obtained with Simplex coding (using L different sequences) with respect to the single-pulse case using L averaged traces (i.e., taking into account similar acquisition time conditions). On the other hand, complementary-correlation Golay codes use pulse sequences with particular correlation properties (Nazarathy et al. 1989). Pulse coding is in this case based on two bipolar sequences Ak and Bk (i.e., containing elements 1’s and C1’s), whose autocorrelation functions have the same correlation peak, but complementary-correlation sidelobes. Therefore, sidelobes cancel out each other when the autocorrelation functions are summed up together, so that f(Ak * Ak ) C (Bk * Bk )g D 2Lı k , where * represents the correlation function, L is the number of bits, and ı k is the Kronecker delta function (Nazarathy et al. 1989) The basic use of Golay codes in Brillouin time-domain sensors requires first to convert

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A Ak and Bk into four unipolar sequences, as uA k D 0:5 .1 C Ak /, uk D 0:5 .1  Ak /, B B uk D 0:5 .1 C Bk /, and uk D 0:5 .1  Bk /. Using these four sequences, then four coded traces are obtained; each one can be denoted as the temporal convolution ˝ between the pulse sequence and the fiber impulse response hk . Thus, finally a decoded trace yk can be obtained by the following decoding calculation:

0

1

0

1

C C B B C C B B C C B B C C B B A A B B C  Ak C B hk ˝ u C  Bk yk D B h ˝ u  h ˝ u  h ˝ u k k k k k k k C B „ ƒ‚ … B „ ƒ‚ … C „ ƒ‚ … C C B B „ ƒ‚ … C C B B @ trace measured @ trace measured trace measured A trace measured A with uA with uBk with uA with uBk k k yk D hk ˝ Ak  Ak C hk ˝ Bk  Bk D hk ˝ f.Ak  Ak / C .Bk  Bk /g D 2Lhk

(64) Equation 64 points out that the coding-decoding process is equivalent to interrogate the fiber impulse response hk with the sum of the autocorrelations of the two sequences. This leads to a coding gain equal to p Cgain D

L ; 2

(65)

One of the fundamental requirements to apply linear pulse coding techniques is to maintain a linear system. This means that the response of the fiber must be the same for each pulse in the sequence, so that a traditional linear decoding process could be used. This restriction implies that the gain coefficient gB (, z) in the sensor response (see Eq. 34) must be the same for all pump pulses. This condition normally cannot be satisfied when using pulses of a few tens of ns (corresponding to a few meters spatial resolution), due to the large cumulated gain induced by the pulse sequences. In case the small-gain approximation cannot be satisfied (i.e., when using a large number of bits, high peak power and long pulses) a linearization of the coded traces is required before decoding. Furthermore, the inertial behavior of the acoustic wave generated by a given pulse typically affects the acoustic field activation produced by other pulses in the code sequence (Soto et al. 2010b). This effect is illustrated in Fig. 36a, which shows a 3 bit sequence f1,0,1g of 10 ns nonreturn-to-zero (NRZ) pulses. It is possible to observe that for the first 10 ns pulse, the acoustic wave amplitude rises from the zero level up to an intermediate level (not fully activated). During the second bit (being a “0”), the activation of the acoustic wave ceases, and its amplitude decays exponentially. However the 10 ns time slot is not long enough for the acoustic wave to completely vanish, and therefore, the activation of the acoustic field during the third bit (bit “1”) rises from an upper level compared to the first one. Consequently, the Brillouin gain induced by the third bit turns out to be larger than the one obtained from the first one. This behavior of the acoustic wave becomes more critical with long pulse sequences, resulting in a

Normalized amplitude (a.u.)

Fig. 36 Pulse coding using (a) NRZ and (b) RZ pulses. Bit patterning effect is eliminated with RZ format (Soto et al. 2010b)

M. A. Soto

Normalized amplitude (a.u.)

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a

NRZ Pump pulses Acoustic wave

sequence

1

1

0

1

0.5

0

0

20

b

40 Time (ns)

60

80

RZ Pump pulses Acoustic wave

1 bit slot 0.5

0

0

50

100 150 Time (ns)

200

250

bit patterning effect that destroys the linearity of the code (Soto et al. 2010b). To solve this issue, bits must be separated by a time interval larger than three to four times the acoustic wave response time  A  12 ns. This time interval (>50 ns) gives enough time to the acoustic wave to be fully damped before the next bit initiates a new acoustic wave activation. This is exemplified in Fig. 36b with a bit sequence f1,1,1g, when using a bit slot of 90 ns. This pulse format is called return-to-zero (RZ) format, and its use is essential to keep the linearity of the code in a Brillouin time-domain sensor (Soto et al. 2010b). Note that so far the description of pulse coding has assumed that bit sequences are sent down the fiber in burst. However, there is also the possibility to spread the code over the entire sensing fiber using cyclic codes, in which the repetition period equals the fiber round-trip time. This approach is based on the use of a singlepulse pattern of length L D 4n  1 (being n an integer number) and having cyclic properties. Examples of this kind of codes are cyclic Simplex codes (Iribas et al. 2017), which have the same coding gain of unipolar Simplex codes, as defined in Eq. 63. Compared to the use of bursts, this format offers important advantages in terms of pump depletion, thus avoiding potential distortions occurring with long unipolar codes and high probe powers.

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Time-Frequency (or Colored) Codes One of the limitations of classical unipolar codes is the reduction of the threshold power for SBS, limiting the peak pump power to levels even below 100 mW for specific spatial resolutions and code lengths. An alternative coding scheme to partially alleviate some of the limitations of (single-frequency) unipolar codes is to use time-frequency codes, also called colored codes (Le Floch et al. 2012; Sauser et al. 2013). In this case, in addition to the intensity coding scheme, pulses with different optical frequencies are used. Compared to single-frequency codes, where coding and decoding processes are independently applied to each scanned frequency, colored codes intrinsically combine intensity-based coding and the frequency scanning required in a Brillouin sensor. This feature imposes one of the main restrictions to this type of codes: the code length L implicitly defines the number of scanned frequencies, which must also be equal to L. This implies that large codes are commonly required to have a sufficiently large spectral scanning range. Note that, since each pulse of the code sequence has a different optical frequency, they activate SBS with different detuning frequencies, resulting in an efficient increment of the SBS power threshold. This allows colored codes to reach the maximum peak power imposed by nonlinear effects such as modulation instability. This kind of codes also mitigates pump depletion issues and nonlocal effects, which might result very significant in single-frequency unipolar codes. This mitigation arises from the fact that the CW probe in this case interacts with pump pulses at different frequencies, reducing the overall pump-probe power transfer when compared to the interrogation of the Brillouin peak frequency using single-frequency codes. There exist different methods to generate colored codes. A possible alternative is to use block-circulant matrix with circulant blocks (BMCB) (Le Floch et al. 2012), following a pattern in which the frequency of each bit slot is defined by the code itself. Under this scheme, a code length L defines L2 pulse sequences, as illustrated in Fig. 37a for the case of L D 3 bits. The coding gain of this

a

b

Fig. 37 (a) 3-bit BMCB-based time-frequency code. (b) 7-bit colored Simplex codes. Colors represent different optical frequencies for the pulses, while the white color is for the absence of the intensity pulse

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p BMCB-based time-frequency code is Cgain D L=2, representing a 1.5 dB higher SNR enhancement when compared to unipolar codes. Alternatively, colored codes can also be constructed following a Simplex coding structure, where simply 1’s and 0’s indicate the presence or absence of an optical pulse, while each time slot is associated with a predefined optical frequency (Sauser et al. 2013), as shown in Fig. 37b for L D 7 bits. Colored Simplex codes have the same gain as the classical p approach, i.e., Cgain D .L C 1/ =2 L.

Bipolar Codes Compared to BOTDR systems, BOTDA sensors offer a unique possibility to extend the types of coding schemes to be used. In particular, combining Brillouin gain and loss processes, the effect of positive (C1’s) and negative (1’s) code elements can be achieved. This allows the implementation of bipolar codes (Soto et al. 2013b), where optical pulses are still generated by intensity modulation, but the positive and negative sign associated with each code element is encoded in the optical frequency of the pulses. The implementation requires a central CW probe signal interacting with a pump signal composed of two optical frequencies, spectrally located around the probe optical frequency, as depicted in Fig. 38a. As also shown in Fig. 38b,

f

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Fig. 38 Bipolar codes in BOTDA. (a) Spectral (b) temporal allocation of C1’s and 1’s (Soto et al. 2013b). (c) Scheme using three probe bands to increase robustness against pump depletion/amplification (Yang et al. 2016)

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bipolar codes are implemented so that “C1” elements are allocated at the upper pump sideband, inducing Brillouin gain on the probe, while “1” elements are placed at the lower pump sideband, inducing Brillouin loss on the probe. Note that, while the single-probe scheme shown in Fig. 38a is conceptually sufficient to describe the implementation of bipolar coding in BOTDA sensors, this scheme is highly affected by depletion of the upper pump sideband and amplification of the lower pump sideband, thus potentially impairing the sensor performance. To increase the robustness against those pump depletion/amplification issues, a scheme using three probe bands is preferred (see Fig. 38c) (Yang et al. 2016), so that the extra probe bands compensate the depletion/amplification that the pump tones experience during propagation. Bipolar codes that have demonstrated significant SNR improvement in BOTDA sensors are bipolar Golay codes (Nazarathy et al. 1989). Compared to unipolar Golay codes, in this case only two pulse sequences are used. These correspond to the original bipolar sequences Ak and Bk defining Golay codes. These sequences result in two coded traces, denoted as hk ˝ Ak and hk ˝ Bk , respectively. The decoding process requires to correlate each coded traces with the respective bipolar sequence launched into the fiber and then sum up the results. Thus, the decoded trace is given as 0

1

0

1

C C B B C C B B C C B B C C B B yk D B hk ˝ Ak C Ak C B hk ˝ Bk C Bk B „ ƒ‚ … C B „ ƒ‚ … C C C B B @ trace measured A @ trace measured A with Ak with Bk

(66)

D hk ˝ f.Ak Ak / C .Bk Bk /g D 2Lhk p The coding gain of bipolar Golay codes is Cgain D L (Soto et al. 2013b). This represents an additional improvement of 3 dB to the SNR of BOTDA traces when compared to unipolar Golay codes (see Eq. 65) of the same length L.

Multiplexing Schemes Time-Division Multiplexing As described in section “Constraints in the Probe Signal of BOTDA Sensors,” to avoid the harmful impact of nonlocal effects and systematic errors, the probe power launched into the fiber must be kept below a certain threshold level. This level is about 5 dBm per sideband in a dual-sideband scheme (see section “Temporal and Spectral Distortion of the Pump Pulse in a Classical Dual-Sideband Scheme”) (Dominguez-Lopez et al. 2016). To boost the probe power with the aim of increasing the measurement SNR and avoid nonlocal effects, a possible solution is to reduce

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the fiber section where pump and probe interact. For this, a long probe pulse can be used instead of the classical CW probe. The probe pulse width defines the interaction length where the pump is depleted (or amplified), while the relative delay between pump and probe pulses defines the interaction region inside the fiber. The Brillouin distribution over the entire fiber is retrieved by measuring different fiber sections by controlling the delay between pump and probe pulses. This technique is known as time-division multiplexing-based BOTDA (Dong et al. 2011). For the implementation, the sensing fiber is divided into several sections of length Li . For the i-th section, an effective interaction length equal to Lieff D Œ1  exp .˛Li / =˛ is defined. In each of these sections the net gain/loss experienced by the pump pulse Gi D gB Psi Lieff =Aeff (where Psi is the probe power entering the i-th section) must be low enough so that pump depletion/excess amplification is avoided. While each of the fiber sections could in principle be defined of the same length, a proper design can be made to maintain the same net gain/loss in each fiber segment. As a consequence, fiber sections with different length could be defined depending on the local probe power. This way, the probe power attenuation along the sensing fiber allows the use of long sections in the first half of the sensing range, where the probe is weak after attenuation, while shorter sections can be defined at far distances, where the probe power is strong. The method allows in principle the use of high probe powers, increasing the measurement SNR without the detrimental impact of nonlocal effects. The only penalty is the longer measurement time required for the acquisition of consecutive time traces over different fiber sections. It must be however pointed out that the use of larger probe powers also allows for the reduction of the number of averages, and therefore big penalties in the overall measurement time are not always expected.

Frequency-Division Multiplexing The purpose of frequency-division multiplexing (Dong et al. 2012) is similar to the one previously described for time-division multiplexing, i.e., to reduce the Brillouin interaction length to avoid pump depletion issues and to allow for an increase in the probe power launched into the fiber with the aim of enhancing the SNR of the time traces. In this case a conventional BOTDA scheme is employed, making use of a pulsed pump and a CW probe signal. However, the sensing fiber is selected such that the entire sensing range is divided into sections having very different Brillouin frequencies. The BFS difference between sections must be ideally larger than the BGS full-width at half-maximum. This way, pump and probe interact only along a reduced fiber length, instead of interacting along the entire sensing fiber. The length of each fiber segment can follow the same rule used for time-domain multiplexing, i.e., short fiber sections can be used at long distances owing to the strong local probe power, while longer fiber sections can be placed at shorter distances where the probe power is lower. The only penalty of this technique is the enlarged pump-probe frequency scanning range that is required to cover the Brillouin gain spectrum of all the fiber

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sections. This typically leads to longer measurement times; however, the acquisition time can be optimized up to some extent by scanning different spectral ranges.

Time and Frequency Pump-Probe Multiplexing There is also another technique based on time and frequency multiplexing that uses very different approach compared to the previous ones (Soto et al. 2014b). The aim in this case is to boost the total power of the pump and probe without activating nonlinear effects. The method makes use of a multifrequency pump pulse and a matching multifrequency CW probe, such that each tone of the pump interacts with a unique tone of the probe, as shown in Fig. 39a. For this, the pump and probe spectral lines must have rigorously the same frequency spacing, allowing for spectrally parallel SBS interactions between single pump-probe pairs. This frequency-division multiplexing approach can be considered as a parallelization of the Brillouin interaction in the fiber. The maximum power of each tone (for both pump and probe) allowed into the fiber is exactly the same as in a conventional BOTDA scheme. Therefore the use of N spectral tones for the pump and probe turns out to be equivalent to N parallel Brillouin interactions, leading to an increase in the sensor response by a factor N. It should be however noted that if high-power pulses at different optical frequencies are simultaneously launched into the fiber, harmful nonlinear processes (such as FWM and MI) will be seeded during propagation. These nonlinear cross-interactions lead to distortions of the measured BOTDA traces, impairing

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Fig. 39 (a) Pump-probe frequency-multiplexing scheme. (b) Time-multiplexing of pump and probe

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significantly the sensor response. To maximize the pulse power and avoid nonlinear cross-interactions among the pump tones, a wavelength-dependent delay must be applied to the N pump pulses, as shown in Fig. 39b. This delay must be similar or longer than the single-pulse duration to secure that pulses do not overlap in time and location anywhere along the fiber (Soto et al. 2014b). However, this gives rise to N similar time-shifted replicas of the BOTDA trace, as illustrated in Fig. 39b. To avoid temporal jamming due to the temporal shift of the pump tones, the N probe signals containing the distributed Brillouin response must be properly rearranged in time before photo-detection by applying a reverse wavelength-dependent delay to the one originally set to the corresponding pump pulse replicas. This temporal realignment of the BOTDA traces (carried by each probe component) is essential to avoid impairing the spatial resolution of the sensor. Although this second temporal shift is preferably performed optically before photo-detection (to allow conventional direct detection), it can also be carried out in the electrical domain or using signal processing after high-frequency photo-detection. The use of this time-frequency multiplexing of pump and probe increases the sensor response and SNR of the measurements. Nevertheless, the scalability of the method is essentially limited by two potential effects (Soto et al. 2014b): (i) fourwave mixing between the probe components and (ii) chromatic dispersion. While the impact of FWM can be considerably reduced using unequally spaced spectral components, special techniques can also be employed to mitigate the impact of chromatic dispersion. Indeed, pump and probe tones can be spectrally allocated following different possible configurations. This may allow for the optimization of the frequency spacing, thus maximizing the number of tones while minimizing the effect of chromatic dispersion. It must be pointed out that the use of a large number of tones, and/or a large spectral separation between lines, can also induce a spectral broadening of the measured BGS, as a result of the wavelength dependence of the BFS. However, a proper allocation of the spectral tones can also minimize this spectral broadening.

Digital Signal Processing Most of the previous techniques for long-range sensing require the modification of the conventional BOTDA scheme. However, there is also the possibility of using signal processing techniques to boost the SNR of the traces measured by classical BOTDR and BOTDA schemes. Hereafter some approaches are presented.

One-Dimensional (1D) Processing The simplest signal processing method to enhance the measurement SNR is the use of digital low-pass filters to reduce noise in the Brillouin sensor data. This approach only bring benefits if the electrical bandwidth of the acquisition system is larger than the optimal one required to achieve a given target spatial resolution. In that case, the measurement bandwidth can be reduced digitally according to the desired resolution. However, in a well-designed system (i.e., when the electrical bandwidth

43 Distributed Brillouin Sensing: Time-Domain Techniques

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Fig. 40 (a) Filter bank representation of the discrete wavelet transform, for a three-level decomposition; h[k] and g[k] are high-pass and low-pass filters, and #2 represents a down-sampling operator by factor 2. (b) Nonlinear thresholding function; continuous line, hard thresholding; dashed line, soft thresholding

matches the spatial resolution requirement), this method only provides a marginal or even null SNR improvement. In such a case, noise reduction using a digital (linear) filter is only effective allowing a degradation of the spatial resolution. To partially overcome this trade-off between spatial resolution and SNR enhancement, nonlinear signal processing methods can be employed. A technique that has shown to be efficient in noise reduction without affecting the spatial resolution of the sensor is the use of a wavelet denoising algorithm (Mallat 1999; Farahani et al. 2012). This unidimensional processing method can be used to eliminate noise from the Brillouin time-domain traces (independently for each scanned frequency), or from the local BGS measured at each fiber location. Wavelet denoising requires three steps. In the first step, discrete wavelet transform (DWT) is used to decompose the signal into multiple frequency bands and obtain coefficients having different levels of precision. Using filter banks (low-pass and high-pass filters) and down-sampling, the signal is decomposed as illustrated in Fig. 40a. For each of these bands, i.e., for each decomposition level, the DWT results in wavelet coefficients wi,j described as (Mallat 1999) Z1 wj;k D 1

1 f .t / j;k .t /dt; with j;k .t / D p 2j



t  k2j 2j

 (67)

where the set of j, k (t) conforms an orthonormal basis formed by dilatation (scaling) and translation (shift) of a mother wavelet (t) and j and k are the scaling

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and translation factors. In the case of the Brillouin sensing data, the DWT results in wavelet coefficients of large amplitude, which are associated with the signal itself (i.e., to the time-domain trace, or local spectrum), and small amplitude coefficients that are associated with noise. As a second step, a nonlinear denoising technique called wavelet thresholding (shrinkage) (Mallat 1999) is then applied to eliminate wavelet coefficients associated with noise, as illustrated in Fig. 40b. To distinguish noise from signal, a threshold level T is defined. Wavelet coefficients having an amplitude below this threshold are associated with noise and are eliminated (set to zero) from the wavelet representation. On the other hand, the amplitude of those coefficients above the threshold (associated with the useful signal) remain unmodified (this is called hard thresholding) or are slightly reduced by an amount equal to T (this is called soft thresholding). One of the main differences between hard and soft thresholding is that soft thresholding usually leads to smoother signals, while hard thresholding better preserve sharp transitions of the signals. Note that shrinkage is applied independently to the different levels of the signal decomposition, and in principle the threshold value can be different for each level, allowing the elimination of noise at any spectral position, while keeping all frequency components of the original signal. There are several methods to define the threshold level; however this is typically determined as a function of the noise standard deviation  . A common p definition is the universal threshold, described as T D  2 log.N /, where N is the number of samples. Note that this is not always an optimal threshold, and sometimes better denoising can be achieve using lower threshold levels (Mallat 1999). As a last step, once wavelet thresholding has been applied, inverse discrete wavelet transform is used to reconstruct a denoised version of the signal using only the remaining high-amplitude wavelet coefficients.

Two-Dimensional (2D) Processing: Image (Denoising) Processing The classical measurement process in a Brillouin distributed sensor results in a map of the local BGS as a function of distance, as shown in Fig. 41a. Data points are commonly stored in a two-dimensional matrix g[z, f ], in which each data point contains the measured Brillouin response at a given position z and frequency offset f, as depicted in Fig. 41b. This two-dimensional data structure makes possible the use of 2D signal processing techniques, such as image denoising algorithms, to eliminate noise from the matrix g[z, f ] by processing each position-frequency pair [z, f ] as a noisy pixel of an image (Soto et al. 2016, 2017). Following this 2D approach, linear and nonlinear image denoising can be efficiently used. Compared to 1D methods, these 2D algorithms are commonly more efficient in terms of noise removal since they can smartly exploit all data patterns existing in the two dimensions of the measured data g[z, f ]. In the case of linear image filters, they commonly use 2D neighborhood (local) operators to reduce noise from noisy pixels in an image. The process of denoising the BOTDA data g[z, f ] can be represented by a two-dimensional convolution resulting in a filtered matrix data gf [z, f ] D h[z, f ] ˝ g[z, f ], where

43 Distributed Brillouin Sensing: Time-Domain Techniques

a 1.5

Gain (%)

Fig. 41 Image denoising for BOTDA data enhancement. (a) 3D map of the BGS measured vs distance. (b) Top view of noisy data. Darker blue tones represent higher SBS gain (Soto et al. 2016)

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h[z, f ] is the bidimensional impulse response of the filter (spatial kernel in the [z, f ] space). Due to the linear and space-invariant properties of this kind of filters, 2D linear image denoising can also be expressed in the Fourier domain as Gf [u, v] D H[u, v]G[u, v], where H[u, v] and G[u, v] are the 2D Fourier transform of h[z, f ] and g[z, f ], respectively. Hence, linear image denoising applied to the Brillouin sensor data can be implemented following two equivalent approaches (Soto et al. 2017): (i) in the spatial domain by processing the data points directly over the [z, f ] space and (ii) in the frequency domain by using direct and inverse 2D Fourier transforms. Linear image filters are simply 2D low-pass filters, and therefore their performance is highly determined by the trade-off between noise removal capability and spatial over-smoothing. Indeed, a linear filter that removes large amount of noise typically blurs and over-smooths the BOTDA data, leading to a loss of highfrequency details commonly associated with the high spatial resolving capability of the sensor. Examples of this kind of filters are the 2D mean filter (also known as moving averaging filter) and the 2D Gaussian filter. To partially overcome the trade-off between spatial smoothing and noise removal, nonlinear image filters can be used. Even though their design and implementation

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are normally more complex than 2D linear filters, they can potentially offer considerably higher SNR enhancement while keeping all high-frequency details of the data. Since they involve the use of a nonlinear function for denoising, which in many cases require space-variant functions, the filtering process cannot be equivalently represented in both space and frequency domain using Fourier analysis. Based on this feature, there exist two independent categories of nonlinear image denoising methods (Soto et al. 2017): (i) pixel-wise techniques, in which a nonlinear denoising operator is applied directly on the [z, f ] space of the Brillouin data, and (ii) transformation-based methods, in which the nonlinear filtering function is applied in a domain representing the frequency content of the data, followed by a process to convert back the denoised data to the original [z, f ] space. Examples of the first category are the median filter, bilateral filter, and nonlocal means (NLM); while transform-based techniques (second category) can be based, for example, on 2D discrete cosine transform or 2D discrete wavelet transform. Although the amount of noise removed with these nonlinear image denoising algorithms highly depends on the data itself (e.g., on the existing patters and redundancies, as well as on the noise level), SNR enhancements of 14 dB have been demonstrated with NLM and wavelet denoising over a 50-km-long classical BOTDA sensor scheme (Soto et al. 2016).

Three-Dimensional (3D) Processing: Video (Denoising) Processing The working principle of Brillouin distributed fiber sensing requires that the entire Brillouin gain information (i.e., including the full frequency scan and averaging) must be acquired in a time shorter than the temporal evolution of the measurand. This feature inherently implies that consecutive BOTDA measurements contain some level of correlated information. A natural extension of the 2D denoising approach described in section “Two-Dimensional (2D) Processing: Image (Denoising) Processing” is to consider also the temporal axis in the processing. This way, 3D processing techniques can be used. Powerful 3D denoising techniques based on video denoising algorithms can therefore be exploited for BOTDA data denoising (Soto et al. 2016). In this case, each BOTDA measurement g[z, f, tk ] is processed as a frame of a video sequence, where tk is the time of the k-th acquisition. Compared to image denoising, this 3D processing approach provides larger noise removal possibilities since video denoising can effectively exploit patterns of information found not only in the [z, f ] space but also in the temporal dimension. Some possible methods for video denoising are the 3D nonlocal means and 3D wavelet denoising. A key feature of video denoising is that the processing takes into account the nonstationary features of the data over time and hence can easily deal with the motion of pixels among different frames. This feature represents a key advantage to minimize delay and over-smoothing of the temporal evolution of the measurand. Using a 3D NLM algorithm, processing ten consecutive frames g[z, f, tk ], an SNR enhancement of 20.7 dB has been demonstrated over a 50km-long BOTDA sensor (2 m spatial resolution, 1.4 dB original SNR and 42 s measurement time) (Soto et al. 2016). It must be noted that, like image denoising

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algorithm, the SNR improvement achieved by video denoising highly depends on the patterns found in the BOTDA measurements, and therefore very different levels of SNR improvement can be expected in real-case scenarios depending on specific sensing conditions.

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S. Le Floch, F. Sauser, M. Llera, M. A. Soto, L. Thévenaz, Colour simplex coding for brillouin distributed sensors, in Proc. SPIE 8794, Fifth European Workshop on Optical Fibre Sensors, 879437 (2013) K. Shimizu, T. Horiguchi, Y. Koyamada, T. Kurashima, Coherent self-heterodyne detection of spontaneously Brillouin-scattered light waves in a single-mode fiber. Opt. Lett. 18(3), 185–187 (1993) M.A. Soto, L. Thévenaz, Modeling and evaluating the performance of Brillouin distributed optical fiber sensors. Opt. Express 21(25), 31347–31366 (2013a) M.A. Soto, S. Le Floch, L. Thévenaz, Bipolar optical pulse coding for performance enhancement in BOTDA sensors. Opt. Express 21(14), 16390–16397 (2013b) M. A. Soto, L. Thévenaz, Towards 1’000’000 resolved points in a distributed optical fibre sensor, in Proc. SPIE 9157, 23rd International Conference on Optical Fibre Sensors, 9157C3 (2014) M.A. Soto, P.K. Sahu, G. Bolognini, F. Di Pasquale, Brillouin-based distributed temperature sensor employing pulse coding. IEEE Sensors J. 8(3), 225–226 (2008) M.A. Soto, G. Bolognini, F. Di Pasquale, L. Thévenaz, Simplex-coded BOTDA fiber sensor with 1 m spatial resolution over a 50 km range. Opt. Lett. 35(2), 259–261 (2010a) M.A. Soto, G. Bolognini, F. Di Pasquale, Analysis of pulse modulation format in coded BOTDA sensors. Opt. Express 18(14), 14878–14892 (2010b) M.A. Soto, G. Bolognini, F. Di Pasquale, Optimization of long-range BOTDA sensors with high resolution using first-order bi-directional Raman amplification. Opt. Express 19(5), 4444–4457 (2011) M.A. Soto, X. Angulo-Vinuesa, S. Martin-Lopez, S.-H. Chin, J.D. Ania-Castañon, P. Corredera, E. Rochat, M. Gonzalez-Herraez, L. Thevenaz, Extending the real remoteness of longrange Brillouin optical time-domain fiber analyzers. J. Lightwave Technol. 32(1), 152–162 (2014a) M.A. Soto, A.L. Ricchiuti, L. Zhang, D. Barrera, S. Sales, L. Thévenaz, Time and frequency pumpprobe multiplexing to enhance the signal response of Brillouin optical time-domain analyzers. Opt. Express 22, 28584–28595 (2014b) M.A. Soto, J.A. Ramírez, L. Thévenaz, Intensifying the response of distributed optical fibre sensors using 2D and 3D image restoration. Nat. Commun. 7, 10870 (2016) M. A. Soto, J. A. Ramírez, L. Thévenaz, Image and video denoising for distributed optical fibre sensors, in Proc. SPIE 10323, 25th International Conference on Optical Fiber Sensors, 103230K (2017) K.D. Souza, Significance of coherent Rayleigh noise in fibre-optic distributed temperature sensing based on spontaneous Brillouin scattering. Meas. Sci. Technol. 17(5), 1065–1069 (2006) K.D. Souza, T.P. Newson, Improvement of signal-to-noise capabilities of a distributed temperature sensor using optical preamplification. Meas. Sci. Technol. 12(7), 952–957 (2001) L. Thévenaz, S.F. Mafang, J. Lin, Effect of pulse depletion in a Brillouin optical time-domain analysis system. Opt. Express 21(12), 14017–14035 (2013) J. Urricelqui, A. Zornoza, M. Sagues, A. Loayssa, Dynamic BOTDA measurements based on Brillouin phase-shift and RF demodulation. Opt. Express 20(24), 26942–26949 (2012) J. Urricelqui, M. Sagues, A. Loayssa, Brillouin optical time-domain analysis sensor assisted by Brillouin distributed amplification of pump pulses. Opt. Express 23, 30448–30458 (2015) A. Voskoboinik, O.F. Yilmaz, A.W. Willner, M. Tur, Sweep-free distributed Brillouin time-domain analyzer (SF-BOTDA). Opt. Express 19, B842–B847 (2011) P.C. Wait, T.P. Newson, Landau Placzek ratio applied to distributed fibre sensing. Opt. Commun. 122, 141–146 (1996a) P.C. Wait, T.P. Newson, Reduction of coherent noise in the Landau Placzek ratio method for distributed fibre optic temperature sensing. Opt. Commun. 131, 285–289 (1996b) P.C. Wait, K.D. Souza, T.P. Newson, A theoretical comparison of spontaneous Raman and Brillouin based fibre optic distributed temperature sensors. Opt. Commun. 144, 17–23 (1997) Z. Yang, M.A. Soto, L. Thévenaz, Increasing robustness of bipolar pulse coding in Brillouin distributed fiber sensors. Opt. Express 24, 586–597 (2016)

Distributed Brillouin Sensing: Frequency-Domain Techniques

44

Aleksander Wosniok

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency-Domain Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement Range and Spatial Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-End-Access Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Advantages of Frequency-Domain Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement Setups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Laser Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Laser Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polarization Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Digital Processing Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The substantial progresses in fiber-optic communications in combination with the increasing economic and political interest in structural health monitoring have led to a commercial establishment of distributed Brillouin sensing. The sensor systems, mainly based on the time-domain techniques, have been successfully implemented in the areas such as pipeline leak detection, geohazard effects, and ground movement detection.

A. Wosniok () 8.6 Fibre Optic Sensors, Federal Institute for Materials Research and Testing (BAM), Berlin, Germany e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 G.-D. Peng (ed.), Handbook of Optical Fibers, https://doi.org/10.1007/978-981-10-7087-7_8

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This chapter introduces a further advancement in the area of Brillouin sensing in the frequency domain. The so-called Brillouin optical frequency-domain analysis (BOFDA) offers crucial perspectives in terms of dynamic range and cost efficiency. The main principle of the frequency-domain approach takes advantage of the reversibility between the time and frequency domain given by a Fourier transform in the analysis of linear systems. The hereby presented overview gives a summary of the benefits and challenges of frequency-domain measurements closely tied to the narrowband recording of the complex transfer function. This function relates the counterpropagating pump and probe laser light along the sensor fiber providing the pulse response of the measurement system by applying the inverse Fourier transform (IFT). The strain or temperature distribution can be then determined from the retrieved Brillouin frequency shift (BFS) profile along the fiber.

Introduction The sensing methods picked out as a central theme of the chapter constitute a direct alternative to well-known time-domain techniques. Contrary to the pulse-based time-domain approaches, in the frequency-domain measurements, only continuous wave light (cw light) of a narrow-linewidth laser is coupled into the sensor fiber. Thereby, all spatially resolved information of the Brillouin gain is included in a sinusoidal modulation of either pump or probe laser light intensity recorded as a complex transfer function of the system. By analyzing the Brillouin backscattered light at different modulation frequencies and at different spectral shifts between pump and probe laser light, the strain and temperature distribution along the whole sensor fiber length can be determined applying the inverse Fourier transform (IFT) to the recorded complex transfer function. The use of the IFT presumes linearity and time invariance of the measured Brillouin gain in the fiber under test. This condition is ensured by using low-intensity laser light coupled into the sensor fiber. The measuring procedure via the frequency-domain transfer function can be seen as a decomposition of the laser pulses into their spectral components. These components are single-frequency harmonic signals corresponding with the frequency values of the performed sinusoidal modulation of laser light intensity. Due to the time-frequency duality given by the Fourier transform, the two concepts of measurement in the time as well as in the frequency domain remain comparable in dynamic range, spatial resolution, and measurement accuracy. However, on the way toward practical realization, frequency-domain techniques offer an essential low-cost potential, as is explained in greater detail below.

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Frequency-Domain Measurements The distributed fiber-optic sensing methods presented in this section use the timefrequency duality of the Fourier transform for linear and time-invariant systems. In contrast to the pulse-based time-domain sensor systems, the Brillouin interaction in the frequency domain takes place exclusively between counterpropagating cw light waves of narrow-linewidth lasers coupled into the sensor fiber. At the same time, a progressive sinusoidal amplitude modulation of either the cw pump or the counterpropagating cw probe light provides the spatially resolved information on the local strains and temperature gradients along the sensor fiber. The progressive sinusoidal amplitude modulation is to be understood as a Fourier pulse decomposition replacing therefore the pulse modulation in time-domain measurements. This section aims to give a detailed description of the theoretical backgrounds on the so-called Brillouin optical frequency-domain analysis (BOFDA) as an alternative to the standard time-domain BOTDA systems. The underlying system theory is only applicable to linear systems. The nonlinearities caused on the one hand by the measuring principle and on the other hand by the stimulated Brillouin effect itself are to be suppressed. These nonlinearities are discussed below, and it is shown how the sensor system can be best linearized. Completing recent advances in the frequency-domain research, a separate subsection is devoted to a reflectometry technique called Brillouin optical frequencydomain reflectometry (BOFDR). The Brillouin optical frequency-domain reflectometry features lower measurement accuracies than BOFDA but allows measurements with one-end access by detecting the spontaneous Brillouin scattering from a sinusoidally modulated pump light.

Theoretical Principles The Brillouin optical frequency-domain analysis BOFDA is based on the measurement of a complex transfer function H that relates the amplitudes and phases of counterpropagating pump and probe light waves along a sensor fiber (Garus et al. 1996, 1997). As shown in Fig. 1 and in Fig. 2, the technique can be implemented by an amplitude modulation of either the probe (loss method) or the pump light (gain method). In both basic configurations, the counterpropagating continuous wave (pump or probe wave) adopts the amplitude modulation at the same frequency fm as the excitation (probe and pump wave, respectively) bearing the system information H( fm ). For the distributed measurement, the modulation frequency is progressively swept by a reference signal of the vector network analyzer VNA with M equidistant frequency steps fm , which transferred into the time domain using an IFT yields the discrete pulse response h(tn, fD ) for a frequency difference fD between the pump and probe waves (Gogolla 2000):

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Fig. 1 BOFDA basic configuration – loss method. VNA vector network analyzer, EOM electrooptic modulator, PD photo diode, FUT fiber under test

Fig. 2 BOFDA basic configuration – gain method. VNA vector network analyzer, EOM electrooptic modulator, PD photo diode, FUT fiber under test

h .tn; fD / D

M 1 1 X H .fm ; fD / exp .2ifm tn / M mD0

(1)

n where: tn D M f und n D 0, 1, 2, : : : , M  1. m By converting the time of detection tn into the location zn ,where the Brillouin 2z n interaction occurred, the substitution tn D nc0 gr in h(tn, fD ) should be performed. Therefore, the spatial pulse response h(zn, fD ) can be easily determined at each location zn in the fiber using the proportionality factor ncgr0 with the vacuum speed of light c0 and the group refractive index of the fiber core ngr .

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The pump and probe waves interact in the fiber by means of stimulated Brillouin scattering (SBS). The Brillouin gain is maximal, if the frequency difference fD D fP  fS between the pump frequency fP and the probe frequency fS is equal to the characteristic Brillouin frequency fB of the fiber. Since the frequency fB depends on the temperature and strain in the fiber, the fully distributed Brillouin gain spectra (BGSs) giving information about temperature gradients and local strain along the fiber under test (FUT) can be recorded as an IFT of the complex transfer function H by varying the frequency difference fD in the range of fB . The loss method (see Fig. 1) corresponds to a configuration with the probe signal (Stokes light) being modulated in amplitude by an electro-optic modulator EOM. The complex baseband transfer function HL ( fm , fD ) is expressed by:

HL .fm ; fD / D

b P .fm ; fD / P exp ŒiˆP .fm ; fD /  iˆS .fm /  b S .fm / P

(2)

b P of the transmitted modulated pump power and the probe with the alternating part P c excitation PS considering the signal phases ˚ P for the pump and ˚ S for the probe wave. The gain method (see Fig. 2) relies on the modulation of pump signal giving the following transfer function HG ( fm , fD ):

HG .fm ; fD / D

b S .fm ; fD / P exp ŒiˆS .fm ; fD /  iˆP .fm /  b P .fm / P

(3)

b S of the transmitted modulated probe power and the pump with the alternating part P b excitationP P . From a metrological point of view, there are any significant differences between the basic configurations presented in Fig. 1 and in Fig. 2. Since the gain of a weak probe signal is easier detectable than the loss of the relatively high pump signal, the following considerations relate only to the gain method. However, the principle validity of the described theory holds also for the loss method. In the gain method, for the pump power PP coupled into the fiber at z D 0, the following relation applies: b P .fm / cos Œ2fm t C P .fm / PP .t; fm / jzD0 D P P C P

(4)

Due to an AC coupling in the photo diode amplifier, both the DC component P P of the pump power and the DC component P S .fm ; fD / of the adopted amplitude modulation of the cw probe signal according to:

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b S .fm ; fD / cos Œ2fm t C S .fm ; fD / PS .t; fm ; fD / jzD0 D P S .fm ; fD / C P (5) are not measured, so that Eq. 3 is valid for the recorded complex transfer function HG . b P .fm /and the phase  P ( fm ) The frequency dependence of the amplitude P traces back to the frequency-specific behavior of hardware components in the measurement setups presented in the section “Measurement Setups.” b S .fm ; fD / is strongly dependent on the modulation frequency The amplitude P fm , basically caused by interference effects due to superposition of modulation signals and by the limited full width at half maximum (FWHM) of the Brillouin gain of about 35 MHz, which ultimately leads to a low-pass behavior. The function  S ( fm , fD ) results from the phase delay time of the modulation components of the probe (Stokes) power along the sensor length. The mentioned physical effects thus b S and  S on the temperature and indicate the dependence of the two quantities P strain conditions of the sensor fiber, which is detected by the complex transfer function HG according to Eq. 3. In order that the IFT of Eq. 3 corresponds to the pulse response given by Eq. 1, the prerequisite of the system linearity must be fulfilled as good as possible. A method to achieve the required quasilinearity is based on the following considerations. There is an exponential (nonlinear) relation between the pump power PP coulped into the fiber with the length L at z D 0 and the probe power PS coupled into the opposite fiber end at z D L given by simplified proportionality: PS .0/  PS .L/ exp . GPP .0//

(6)

with G representing a Brillouin gain parameter. The accurate composition of G can be neglected here for the sake of simplicity. Putting expression (5) for modulated probe signal into Eq. 6, the following relation can be taken as a second-order Taylor polynomial approximation:  2 1 PS .0/ PS .L/ 1 C GP P mP cos .2fm t / C GP P mP 4  2 1 C GP P mP cos .22fm t / 4

(7)

where the last term is a harmonic wave to be suppressed in the adopted modulation of the probe signal. Particularly, this can be achieved both by a low pump power PP (0) and by its PP (less than 1). The value m can be primarily low degree of modulation m D b P

PP

P

determined selecting an appropriate amplification of the reference output signal of b P ). the network analyzer VNA (setting of P

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Measurement Range and Spatial Resolution The feasible BOFDA measurement range is specified on the one hand mathematically by the properties of the discrete inverse Fourier transform (DIFT) and on the other hand physically by the optical attenuation along the sensor fiber. According to Bracewell (1999), a DIFT generally features a periodic continuation of the pulse response h(t). Therefore, h(t) represents a periodic function whose period PF has its seeds in the discretization of the scanned transfer function H( fm ) and can thus be expressed by the following relation:

PF D

1 M  D  max fm fm  fmmin

(8)

where fmmax and fmmin are the maximum and minimum modulation frequencies, respectively. If the length-dependent scan time of the sensor fiber is greater than PF ,the adjacent pulse responses overlap, which leads to a distortion of the measurement signal. To avoid this, the maximum sensor length zmax must not exceed the following value: zmax D

c0 M c0   D 2ngr fm 2ngr fmmax  fmmin

(9)

In practice, zmax is additionally limited by the occurrence of undesired optical losses both due to critical sensor bending (microbending and macrobending effects in optical fibers) and due to bad splice and plug connections. The frequency-limited measurement of H can be treated as a multiplication of 0 a transfer function H recorded over an infinite frequency range with a normalized rectangular function rect( fm ) going from fmmin to fmmax . Accordingly, this means the following convolution in the time domain by applying IFT F 1 : h.t / D h0 .t / F 1 Œrect .fm /

(10)

which results under the assumptions fmmax  fmmin and fmmin ! 0 in:   h.t / D fmmax  h0 .t /  sinc 2fmmax t Consequently, the width @t of the main peak of the sinc function sinc.t / D sets the maximum spatial resolution @z of BOFDA:

(11) sin.t/ t

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@z D

c0 c0 @t D 2n 2nfmmax

(12)

In order to achieve the spatial resolution of 1 m, a stepwise amplitude modulation of the optical pump wave with the modulation frequencies fm up to 100 MHz is therefore required. However, the crucial limitation of the spatial resolution @z is due to the lowpass behavior of the modulated optical signals mentioned above. This manifests itself as a degradation of signal-to-noise ratio (SNR) of the recorded BGSs. In other words, the Brillouin gain spectra become narrower and noisier as the values fm increase. Figure 3 shows that the amplitude modulation of the pump wave with the modulation frequencies fm up to 100 MHz (spatial resolution of 1 m) leads to a good approximation of Lorentz-shaped profiles of the BGSs. The spatial resolution of 1 m proves to be a reasonable limitation to meet measurement accuracy requirements at acceptable measurement times whose specifics are explained closer separately in the next subsection. The spatial resolution @z describes the ability of a BOFDA system to distinguish small-sized events along a sensor fiber with distinct temperature-strain state (different BFSs). The occurrence of two events within the lengths below @z leads to a distortion of the Brillouin spectra depending on the BFS difference between the separate events related to the FWHM of the Brillouin gain. Figure 4 illustrates two different cases of two separate events occurring below the spatial resolution limit, what simultaneously impairs the measurement accuracy in determining distributed BFS values.

Measurement Time The measurement time in BOFDA method is conditioned by a narrowband filtering of the measurement signals using VNA and by propagation time of the light signals in the sensor fiber. This means, for example, 0.3 ms for the filter bandwidth of 3 kHz and 0.1 ms for a typical sensor length of 10 km. Since according to Eqs. 9 and 12 104 frequency modulation steps are needed to achieve the spatial resolution of 1 m, the required measurement time is approximately 4 s for each setting fD D fP  fS . The frequency range fDmax  fDmin to be swept in order to record spatially resolved BGSs carrying information about strain and temperature distribution along the fiber is dictated mainly by the expected strain level. As shown in Fig. 3, this means a range of about 300 MHz for strain values of about 0.5% (5000 "). If it is taken into account that in case of a precise measurement a frequency step fD of 1 MHz should be set, the total measurement time would be around 20 min. Therefore, in consideration of relatively long total measurement times in BOFDA, a compromise between a sensor length, measurement accuracy, and spatial resolution, which are over fD , fm ,and fmmax the basis for the total measurement time, is to be found in most applications.

Fig. 3 Spatially resolved Brillouin gain spectra of an inhomogeneously strained fiber (160 m) measured at fmmax D 100 MHz. Left: 3D measurement diagram (spatial resolution: 1 m). Right: Brillouin gain spectrum at z D 20 m

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Fig. 4 Brillouin gain spectra for two different events (different BFSs) occurring below the spatial resolution of 1 m. Left: BFS difference greater than the FWHM of the Brillouin gain. Right: BFS difference smaller than the FWHM of the Brillouin gain

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As explained above, the data acquisition time using VNA is decisively limited by the setting time of its analog filters, which is inversely proportional to the bandwidth of the filters. However, a measurement of the transfer function via VNA is not absolutely necessary. A VNA is even clearly overdesigned for recording the single-scattering parameter S21 corresponding to the transfer function H( fm ) in the BOFDA setups presented in section “Measurement Setups.” In Nöther (2010), a concept of a digital BOFDA system is presented in detail. Since here the complete digital processing can be performed off-line, the digital BOFDA features a significant advantage of data acquisition time reduction (Wosniok et al. 2009). A comprehensive introduction to the digital concept of BOFDA is provided in the section “Digital Processing Technique.”

Measurement Accuracy All local strains " and temperature changes T can be determined from the distributed BGSs recorded along a whole sensor fiber as already shown in Fig. 3 left. Each individual spectrum of a standard single-mode fiber (SMF), used commonly as a distributed sensor, features a Lorentz profile (see Fig. 3 right) giving an explicit value fB of BFS defined as a frequency at a maximum of BGS. The value fB is in turn sensitive to both longitudinal strain and temperature as follows:   ı fB .T; ©/ D CT  .T  TR / C C©  © C fB0 TR D 20 C; 0

(13)

in which CT and C" describe the Brillouin thermal and strain coefficients, respectively. The strain and temperature accuracy is thus directly related to the issue of uncertainty in BFS determination. The measurement of fB can be negatively affected on the one hand by Brillouin gain weakening along the fiber and on the other hand by distortion of Brillouin gain spectra themselves. The Brillouin gain weakening can be caused both by extrinsic factors such as microbending and macrobending effects, bad splice, or plug connections and by measurement-related factors such as polarization setting or choice of frequency step fD . Apart from physical reasons of too high fmmax value described above, the distortion of Brillouin gain spectra finds its origin in discrete inverse Fourier transform artifacts. Due to the metrological relevance of SBS polarization effects, the polarization adjustment is separately presented in the section “Polarization Effects.” In general, the Brillouin gain depends strongly on polarization conditions of pump and probe wave represented by a polarization factor  . The polarization factor  can have values between 0 and 1.  D 0 corresponds to the situation where the polarization vectors of the two counterpropagating waves (pump and probe) are perpendicular to each other, so that no interaction takes place (complete failure of SBS).  D 1 represents a maximal Brillouin amplification achieved by parallel orientation of polarization vectors of pump and probe wave. Therefore, a misalignment of

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polarization vectors   0 results in weak-gained Brillouin spectra. In order to avoid this polarization effect, which can occur on several points along the sensor fiber due to the ever-present birefringence, the critical polarization states can be successfully controlled by polarization controllers and scramblers as described in “Digital Processing Technique.” The distortion of Brillouin gain spectra referring to the discrete inverse Fourier transform artifacts can be ascribed to two effects: overlapping and leakage effect (Gogolla 2000). The former effect defines the measurement range characterized in “Measurement Range and Spatial Resolution.” The overlapping means that a mismatch set value fm and the real sensor length lead to an overlap of high-order periodic continuation of the pulse response to the first-order pulse response h(t) which contains the measurement data to be determined. In this way, an additional defective signal component results from this overlap distorting the recorded BGSs ultimately. This negative effect can easily be avoided by choosing a suitable frequency step fm which arises from Eq. 9 for a given sensor length zmax . The frequency-limited detection of the transfer function H also manifests itself by the leakage effect inducing overshoots in the pulse response h(t). The ideal case of detecting the measurement signals in the unlimited modulation frequency range corresponds to a Dirac pulse ı(t) in the time domain. According to Eq. 11, the real case of the frequency-limited measurement leads in turn to a convolution of the 0 ideal pulse response h (t) with a sinc function in the time domain. As shown in Fig. 5 with IFT[rect( fm ) ], the sinc function features a relatively narrow width of the main lobe and high side lobes. The occurrence of the mentioned overshoots is attributed to the high side lobes of the sinc function. In order to suppress this negative effect, the measured transfer function H( fm ) is to be multiplied by a suitable window function, which additionally results in a corresponding convolution in the time domain. However, the simultaneous requirements for a narrow width of the main lobe (influence on the spatial resolution) and the lowest possible side lobes (influence on the measurement accuracy) of the window function can practically not be fulfilled. A large number of window functions (Riemann, Bohman, HanningPoisson, Hamming, Kaiser-Bessel, etc.) are available in the literature (Harris 1978) which offers specific advantages for different tasks in the signal analysis. In the method of BOFDA, the Kaiser-Bessel functions, shown in Fig. 5, with the selectable window parameter ˇ in the range of 3 to 4 proves to be the most suitable. As a result of the above-described optimization steps, BOFDA setups can provide measurement accuracies in the range of 0.5 MHz using SMFs as a distributed sensor, which corresponds to 0.5 ı C for temperature measurement and 10 " for monitoring of mechanical deformation, respectively. Thus, BOTDA and BOFDA are comparable in terms of achievable measurement accuracies. Compared to the Lorentzian BGS of the SMFs, the BGSs of the multimode silica fibers (MMFs) proved to be detectably wider and little distorted (Minardo et al. 2014). This behavior slightly impairs the BFS determination and thus the measurement accuracy in MMFs. With respect to issues of accuracy of the BFS determination, unwanted spectrum distortion can be traced back to the SBS caused

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Fig. 5 Inverse Fourier transform of a normalized rectangular function and the normalized KaiserBessel function for different ˇ parameters. The upper cutoff frequency of all calculated window functions is 25 MHz

by higher-order acoustic modes supported by corresponding high-order optical modes in a MMF. In some real applications, an access to both sensor fiber ends required for BOFDA operation can be considerably hindered or even impossible. A way out of such a problem in practice is given by a one-end accessibility using Brillouin optical frequency-domain reflectometry technique (BOFDR) presented below. By detecting the spontaneous Brillouin scattering, BOFDR measurements remain in fact free from signal distortion related to acoustic wave modulation at high frequencies fm (Minardo et al. 2016) characteristic for BOFDA lowpass behavior. However, the measurement accuracy of BOFDR, in the range of a few MHz at best, is fundamentally limited by the nature of the spontaneous Brillouin scattering providing substantially lower SNRs (Minardo et al. 2016) and wider BGSs (Yaniay et al. 2002) compared to BOFDA based on SBS. With a focus on application-relevant high requirements regarding measurement precision, individual optimization steps of BOFDA approaches are discussed in detail in section “Measurement Setups.”

One-End-Access Measurements Brillouin sensing systems are usually based on SBS generated by coupling frequency-shifted laser light beams into both ends of a sensor fiber. Compared to such Brillouin analyzers (BOFDA systems), a reflectometry approach proposed

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in Minardo et al. (2016) allows for a Brillouin configuration, in which a light beam is injected only into one fiber end. As described in the section above, the advantage of one-end accessibility in BOFDR is realized at the expense of the measurement accuracy derived from the character of the spontaneous Brillouin scattering used in BOFDR. In particular, the measurement accuracy related to estimation error of BFS decreases with the square root of the FWHM of the measured BGS (Soto and Thévenaz 2013). With the value of about 90 MHz (Yaniay et al. 2002), BGS in case of the spontaneous Brillouin scattering is considerably wider than its 35-MHz-wide SBS counterpart. Moreover, the FWHM of the measured BGS increases with the bandwidth of the electrical band-pass filter BPF used in the simplified BOFDR setup shown in Fig. 6 according to Minardo et al. (2016). Since the spatial resolution of BOFDR given by Eq. 12, in analogy to BOFDA, is also linked to the BPF bandwidth, a trade-off must be made between the measurement accuracy and the spatial resolution in choosing BPF bandwidth. The choice of BPF bandwidth in the range of 100 MHz corresponds to a spatial resolution of 1 m, and it allows for reasonable SNR values. Similar to BOFDA, the operation of BOFDR presented in Fig. 6 is based on amplitude modulation of the pump wave by sweeping the modulation frequency fm . Due to the spontaneous Brillouin scattering in the sensor fiber, the Brillouin backscattered signal comprises both Stokes and anti-Stokes component. The letter can be filtered out by the use of a narrowband fiber Bragg grating (FBG) after propagating back to the fiber input, while the Stokes component is converted into an electrical signal by a photodetector PD at the beat frequency fB . After conversion to the electrical domain, the electrical signal of PD at the value fB of BFS is frequencydownshifted to the intermediate frequency (IF) band, by mixing it with the tunable

Fig. 6 BOFDR basic configuration. VNA vector network analyzer, EOM electro-optic modulator, PD photo diode, SG signal generator, BPF band-pass filter, ED envelope detector, FUT fiber under test

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signal provided by a microwave signal generator SG. After the following filtering and amplification process by the use of a BPF, the baseband signal is recovered by passing the IF signal through an envelope detector ED connected to the VNA. In order to record spatially resolved BGSs carrying information about strain and temperature distribution along the fiber, the frequency of SG is swept in the range of fB according to expected temperature gradients and local strain values along the sensor fiber. All frequency-domain approaches feature a progressive sinusoidal amplitude modulation of the cw laser light coupled into a fiber. This also leads to a joint conclusion about the measurement range mathematically expressed by Eq. 9. In fact, the BOFDR technique proves to be indeed inherently more vulnerable to additional optical losses due to bending effects along the sensor fiber than its BOFDA counterpart described in further detail below.

Advantages of Frequency-Domain Technique The frequency-domain analysis methods for distributed strain and temperature measurement along even several tens of kilometers offer some benefits compared to the standard BOTDA concept. An important asset of BOFDA can be directly derived from the measurement principle based on a narrow-bandwidth operation. The BOFDA measurement bandwidth is specified by a narrowband filter of VNA, typically in the kHz range around the modulation frequency fm , whereas the BOTDA measurement bandwidth of at least 100 MHz is necessary to resolve pulses of several nanoseconds. Since SNR is inversely proportional to the measurement bandwidth, BOFDA enables a significant improvement of SNR up to 50 dB. However, this statement should not be overestimated by considering that, relating to pump depletion properties, significantly higher laser powers directly contributing to SNR improvement can be used in BOTDA (Gogolla 2000). Moreover, for noise suppression in BOTDA, a few thousand individual measurements are averaged (Bao et al. 1995). Another advantage of BOFDA is that this technique overcomes the need for use of expensive high-speed electronics for the purpose of generation and detection of short laser pulses, which in turn reduces system costs. From the theoretical point of view, the narrow-bandwidth operation in case of BOFDA should result in relatively long measurement time. The associated problem of the long setting times of analog filters in a VNA’s internal circuitry discussed in the section “Measurement Time” can be solved by adopting of a digital setup approach as described in detail in the section “Digital Processing Technique.” Such digitization combines two benefits: on the one hand, it reduces the measurement time, and on the other hand, it offers a decisive cost advantage by replacing an expensive VNA with a digital circuit. In addition, the existing low-cost potential of BOFDA is also intensified by the use of low-power lasers which are operated below the pump threshold (prerequisite of the system linearity).

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Measurement Setups The measuring procedures of BOFDA follow an accurate algorithmic pattern of generation of a progressive sinusoidal amplitude modulation of the pump wave (gain method) by sweeping the modulation frequency fm for a selected constant laser frequency difference fD of counterpropagating pump and probe signals. The values of the transfer function HG ( fm , fD ) recorded by the sweep provide a complete signal trace h(z, fD ) over the whole fiber length using DIFT. Repeating this sweeping process of fm for each value of fD set in the range of characteristic Brillouin frequency shift fB , three-dimensional images of BGSs can be determined as already shown in Fig. 3 left. This section focuses on the advancements of BOFDA setups by presenting two hardware solutions for the realization of the fD scanning to be performed for the purpose of recording distributed BGSs along a sensor fiber. The starting point for implementation of the measurement algorithm is given here by a two-laser setup. As a result of adoption of the idea of modulating the laser’s amplitude with an RF signal with a frequency in the range of fB corresponding the sideband technique proposed in Nikles et al. (1997), the relaxation and setting time of a control loop to drive two separate lasers can be reduced using only one laser source in the BOFDA setup. This section also gives information on crucial optimization steps for increasing measurement accuracy (polarization adjustment) and improving the measurement time with a simultaneous system cost reduction using digital processing.

Two-Laser Configuration The first investigation on the optical frequency-domain technique referred to a distributed reflectometry method for measurement of linear backscattering (Ghafoori-Shiraz and Okoshi 1986). This approach was adopted in the 1990s for Brillouin sensing (Garus et al. 1997; Gogolla 2000; Krebber 2001). Within this research, numerical models of SBS for BOFDA and laboratory measurements along 11-km-long optical fiber with a spatial resolution of 1 m were presented. The BOFDA setup described here in this section is based on advancements achieved by investigations within the program “Risk Management of Extreme Flood Events (RIMAX)” of the German Federal Ministry of Education and Research (2005– 2009) offering new perspectives in terms of dynamic range and measurement accuracy. Figure 7 shows a complete BOFDA laboratory setup based on two separate Nd:YAG lasers used as frequency-shifted pump and probe (Stokes) signal sources. In the measurement setup above, two separate functional blocks can be identified. The first functional group serves to control and adjust the frequency differences fD D fP  fS , at which the transfer functions are recorded by a VNA. This control loop is composed of a photodetector PD (upper left) with a bandwidth of 25 GHz, an electrical spectrum analyzer SA for the detection of the current beat frequency

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Fig. 7 Basic two-laser configuration of BOFDA. VNA vector network analyzer, EOM electrooptic modulator, PD photo diode, DAC digital-to-analog converter, SA electrical spectrum analyzer, PC/PS polarization controller/polarization scrambler, FUT fiber under test

fD , and a digital-to-analog converter DAC, directly developed for voltage control of the pump laser frequency fP . Both Nd:YAG lasers, emitting at the wavelength of 1.319 m and with a bandwidth of 5 kHz, are tunable by means of electrical signals. Such frequency tuning is achieved by altering the emission wavelength of the crystal resonator in two ways simultaneously. First, rough tuning in a range of 30 GHz is carried out by a peltier element acting as a thermoelectric cooler (TEC) over a voltage UT attached to the laser crystal and characterized by a coefficient of 3.8 GHz/V. Second, fast frequency tuning over a voltage UP for fine adjustments in a range of 30 MHz is realized by means of a piezoelectric element mounted on the laser crystal and characterized by a coefficient of 1.0 MHz/V. The spectrum analyzer communicates with a control computer over a GPIB interface. As presented in Nöther et al. (2008a, b) and described in detail in Nöther (2010), the two required voltage values UT und UP are computed by a special algorithmic calculation and sent to a digital-to-analog converter over a serial port RS-232. If the current laser frequency difference fD controlled by the spectrum analyzer meets the accuracy requirements of fD (here 150 kHz) for fD , then the measurement of the transfer function is being continued by the VNA for the set frequency difference fD . As shown in Fig. 8 (Wosniok 2013), the initial operating point fD (UTP , UTS ) should be favorably set centrally between two adjacent laser mode hops around the natural BFS of 12.8 GHz at the wavelength of 1.319 m. It is defined by a rough tuning via TEC of the pump laser (control voltage UTP ) or of the probe (Stokes) laser (control voltage UTS ). Figure 8 depicts the setting of the initial operating point by setting UTP , when the rough tuning of the Stokes laser is switched off (UTS D 0). The vector network analyzer VNA is the core element of the second block in the BOFDA setup presented in Fig. 7. The VNA is used for the purpose of recording

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Fig. 8 Setting the laser operating point between mode hops. UTP voltage on TEC of the pump laser, UTS voltage on TEC of the probe (Stokes) laser

individual measurement signals H( fm , fD ) and providing RF input of the EOM with an electrical modulation signal at the adjustable frequency fm at the same time. The electrical modulation signal generated by the internal VNA oscillator serves also directly as the reference signal containing information about amplitude and phase of the system input (pump signal). The electrical information about system output (probe signal) needed for calculating the scattering parameter S21 – transfer function H( fm , fD ) – is received using a photodetector with a bandwidth of 1 GHz, whose AC-coupled output is connected to the VNA as shown in Fig. 7. The polarization controller in the pump path serves to allow for the best possible state of linear polarization required to maximize the extinction ratio of the EOM. The polarization controller or scrambler in the probe path improves the average Brillouin gain along the sensor fiber. The generation of the beat frequency fP  fS by means of two laser sources according to Fig. 7 is hampered by the still existing drift of the laser frequency, which impairs the measurement times by the necessary adjustment of the pump laser frequency fP in order to control the frequency difference fD (adverse relaxation and setting time of a control loop). In general, the reasons for the frequency fluctuations are the digital-to-analog converter noise, temperature changes in laser resonator, or vibrations. The environmental parameters such as air temperature, humidity, and pressure can also increase the drifting in a slow way. The drifting problem can be circumvented by applying the sideband technique described below, which downsizes the BOFDA setup from two laser sources to one.

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One-Laser Configuration In the implementation of the sideband technique (Nikles et al. 1997) for BOFDA, the cost-effectiveness of the measurement system can be additionally enhanced by the use of a convenient laser diode widely adopted in optical communication applications. Figure 9 shows a BOFDA laboratory setup built up on one single semiconductor distributed feedback laser diode (DFB laser diode) operating at the wavelength of 1.55 m and with a bandwidth of 300 kHz. Since the value fB of the BFS is inverse proportional to the wavelength of the pump wave, the natural BFS changes here from 12.8 GHz achieved by Nd:YAG laser source at the wavelength of 1.319 m to 10.9 GHz. This inverse proportionality also B applies to the thermal CT D ıf and strain C© D ıfı©B coefficients appearing in Eq. 13 ıT as summarized in Table 1 for a standard single-mode fiber SMF-28e.

Fig. 9 Basic one-laser configuration of BOFDA using the sideband technique. VNA vector network analyzer, EOM electro-optic modulator, PD photo diode, SG signal generator, PC/PS polarization controller/polarization scrambler Table 1 Thermal and strain coefficients of the SMF-28e in the second (P D 1.319 m) and third (P D 1.55 m) near-infrared spectral windows Wavelength P [m] 1.319 1.55

CT [MHz/ı C] 1.3 1.1

C" [MHz/103 "] 56 48

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Compared to Fig. 7, the one-laser configuration presented in Fig. 9 requires an additional EOM to modulate the optical power in amplitude using a signal generator SG at a frequency fD D fP  fS in the range of fB ( 10.9 GHz). Such amplitude modulation gives rise to new sidebands in the optical spectrum. The lower sideband at the frequency fP  fD D fS acts as the frequency-downshifted probe signal playing the same role as the Stokes laser in the two-laser configuration. The upper sideband at the frequency fP C fD D fS C 2fD amplifies the counterpropagating pump signal at the frequency fP in the sensor fiber. Since the intensity of the upper sideband is very low compared to the pump intensity, the intensity increase at fP due to this interaction is minor and can be neglected. On the contrary, the upper sideband, here in the role of a pump signal to the modulated counterpropagating pump wave at the frequency fP , is significantly depleted due to the high intensity differences of the two interacting signals. By setting a DC voltage of the additional EOM to a minimum optical transmission, so that no light is transmitted in absence of the amplitude modulation, the carrier at the frequency fP can be suppressed. As a result, only the interaction between the modulated pump wave and the counterpropagating lower sideband as the probe signal is relevant from the viewpoint of the measurement principle. The two EOMs (called also Mach-Zehnder modulators) are operated with different DC bias voltages to adjust the operating points of the modulators in accordance with their separate requirements. As mentioned above, a DC voltage of the EOM should be set to a minimum optical transmission to suppress the carrier, whereas the modulation of the pump wave is carried out in the linear range, i.e., in the middle between the maximal and minimal transmission. Since the signal quality of the spatially resolved BGSs recorded using the described one-laser configuration can be degraded by nonlinear effects that arise from undesired interaction between the carrier and the sidebands, the DC voltage is to be adjust to a small drift of the modulator transfer function during the measurement process. Such a drift can result in a change of the optical signal, especially in case of the additional EOM in the probe path. Here, the modulated optical signal can be seriously affected by a rebuilding of the suppressed carrier if the DC voltage is not corrected.

Polarization Effects The dependence of the Brillouin gain on the states of polarization (SOPs) of the two counterpropagating pump and probe waves can generally be summarized into a polarization factor  described in the section “Measurement Accuracy.” If  (z) assumes values close to zero along the fiber, a significant deterioration of the accuracy in the determination of BFS values fB is to be expected due to weak measuring signals. As already mentioned in the first section, even “dead zones” can occur at each individual measurement point with  D 0, where no SBS is observed. To eliminate these unacceptable sensor states,  (z)  0 must be avoided at any location z along the fiber.

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In conventional optical fibers used in telecommunication, the polarization state of a light wave propagating through the fiber changes continuously due to the inherent birefringence of the fiber (Nöther et al. 2008b). The stochastic nature of the physical effect of the birefringence makes the control of the polarization states along the measured section so difficult. Already during fiber manufacturing process, slight and random deviations from an ideal circular cross-section of the fiber core occur, which results in minor differences in the refractive index (nx > ny ) along the two main axes x (slow axis) and y (fast axis) causing the birefringence B: B D nx  ny

(14)

Local birefringence is additionally affected by mechanical stress caused by different dopant concentration in the fiber core and cladding as well as by torsion applied to the fiber during the manufacturing process (elasto-optic changes in effective refractive indices). In spite of the random character of the linear birefringence formation, some individual key factors influencing the birefringence are quantitatively described in Rashleigh (1983). According to Rashleigh (1983), also external effects, such as compressive or tensile forces, contribute to the birefringence. For sensor technology purposes, a very important finding is the fact that a macrobending of optical fibers also results in variation of the material birefringence properties as: B

 r 2

(15)

R

where r and R are the fiber and bending radius, respectively. Conditioned by the way of sensor installation, the random fluctuations of the state of polarization as well as the optical losses can be restricted. This in turn can increase the measuring accuracy. Due to the birefringence B given by Eq. (14), a light wave with the wavelength  injected into the sensor fiber will adopt all possible states of polarization within the beat length LB defined as: LB D

 B

(16)

In low-birefringence fibers, like standard telecommunication single-mode silica optical fibers typically used as distributed sensors, the beat length is below 100 m. According to Foschini and Poole (1991), the development of light’s SOPs in this type of fiber can be described as a three-dimensional Brownian motion. Since the BOFDA measurement ranges of several kilometers largely exceed the beat length LB , the SOPs are evenly distributed over the Poincaré sphere with good  approximation. This leads to the SBS mixing efficiencies of  D 23 for coinciding 

polarizations of counterpropagating pump and probe (Stokes) waves and  D 13 for orthogonal polarizations. Therefore, even in the case of a simple rotation of one of

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Fig. 10 Polarization matching of the Brillouin gain distribution along a SMF-28e for linearly polarized pump and probe light

the two polarization vectors, an average Brillouin gain of the measurement signals backscattered in the SBS process can be enhanced by factor 2. Such a polarization adjustment of the Brillouin gain along a standard SMF-28e fiber is shown in Fig. 10. The relative SOPs of the interacting light waves have been matched here by turning the polarization direction of the linearly polarized Stokes wave (probe signal) by means of a polarization controller (PC/PS in the probe path used as a PC) in the BOFDA setup according to Fig. 7. Figure 10 illustrates significant fluctuations in the backscattered distributed Brillouin signals occur for measurements with predefined SOPs of the pump and the probe light coupled into the two opposite fiber ends. In order to obtain a homogeneous Brillouin gain over the entire fiber length, two approaches to overcome the polarization-related signal fluctuations are presented in this section. The first approach is based on an averaging procedure of the BOFDA measurement at a fixed number of SOPs that are distributed over the Poincaré sphere. The violet curve in Fig. 11 represents the result of such an averaging of two signal traces recorded at orthogonal SOPs and at the frequency difference fP  fS D fB . This averaged signal trace shows that an averaging over more different SOPs is necessary to smooth the Brillouin trace over the whole length of the sensor fiber which automatically increases the measurement time. The second approach which can be used to cope with the polarization-related fluctuations is to scramble one of the two SOPs of the interacting light waves. In

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Fig. 11 Polarization-related fluctuations of the Brillouin gain distribution along a 200-m-long SMF-28e measured by the BOFDA system. All measurements were performed at a constant pump and Stokes optical power, VNA’s bandwidth of 3 kHz, and at constant temperature conditions of the unstained FUT

this method, the polarization direction of either the pump or the probe (Stokes) wave is rapidly rotated by means of a polarization scrambler PS before coupling into the fiber, whereas the counterpropagating light wave is injected into the fiber at the opposite fiber end as a linearly polarized wave. Figure 11 points out that the polarization-related fluctuations of the spatially resolved measurement signals can be effectively suppressed by the choice of higher rotational frequencies fR of the polarization rotation. The enhancement of the rotational frequency fR from 500 Hz to 700 kHz presented in Fig. 11 results in a visible increase of the polarization independency of the measured BOFDA signals, which is reflected in the improvement of the corresponding standard deviation from 5.2% to 2.9%. In general, in order to obtain a low polarization disturbance, the following condition must be fulfilled (Gogolla 2000): fR > 10 BF

(17)

Narrowing of the filter bandwidth BF within the VNA to eliminate polarization fluctuations leads directly to an increase of the measurement time (the settling time of the VNA’s analog filters is inversely proportional to their bandwidth – see section “Measurement Time”). When seen from this aspect, the increase of fR seems to be the most efficient for further elimination of the polarization fluctuations. At the same

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Fig. 12 Digital optical signal processing for BOFDA. DDS digital data synthesis, ADC analog-todigital converter, DAC digital-to-analog converter, LPG low-pass filter, VGA variable gain amplifier

time, compared to the common scrambling modules, the electro-optic polarization scramblers offer substantially higher rotational frequencies fR up to about 20 GHz.

Digital Processing Technique The central idea of the digital concept is to replace the overdesigned and expensive VNA with a digital setup recording the complex transfer function. The general concept of a cost-effective digital signal processing is given in Fig. 12 (Wosniok et al. 2009). Similarly to the analog solutions discussed above, the amplitude modulation of the pump signal is carried out by an electro-optic modulator; however, the modulation frequency fm is generated by a digital synthetizer using a direct data synthesis (DDS) circuit (Nöther 2010). This modulation signal is used both to drive the electro-optic modulator and to be directly converted back into the digital domain giving a reference signal for calculation of the transfer function H. The signal to be measured with the information about the Brillouin gain along the sensor fiber is detected by a photo diode PD, limited in its bandwidth by an anti-aliasing low-pass filter LPF, and finally digitalized by an analog-to-digital converter ADC. In this way, the digital data can be transferred to a computer where a definite off-line fast Fourier transform (FFT) is to be performed. Since DDC and ADCs are standard electronic devices, the cost of the measurement components are significantly lower for the digital implementation. The off-line signal processing enables fast measurements at a high dynamic range.

Conclusion This chapter provides a compact overview of fundamental principles of the Brillouin frequency-domain techniques for structural health monitoring applications. Starting from the theoretical backgrounds on measurement in the frequency domain, the

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BOFDA approaches using the one- and the two-laser configurations were presented in detail. With the focus on real-life temperature and strain measurement applications, the advantages and limitations of distributed Brillouin sensing based on both the highly accurate frequency-domain analysis and the practically advantageous one-end-access reflectometry were discussed.

References X. Bao, J. Dhliwayo, N. Heron, D.J. Webb, D.A. Jackson, J. Lightwave Technol. 13(7), 1340–1348 (1995) R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1999) G.J. Foschini, C.D. Poole, J. Lightwave Technol. 9(11), 1439–1456 (1991) D. Garus, K. Krebber, F. Schliep, T. Gogolla, Opt. Lett. 21(17), 1402–1404 (1996) D. Garus, T. Gogolla, K. Krebber, F. Schliep, J. Lightwave Technol. 15(4), 654–662 (1997) H. Ghafoori-Shiraz, T. Okoshi, J. Lightwave Technol. 4(3), 316–322 (1986) T. Gogolla, Theoretische Untersuchung der Brillouin-Wechselwirkung in Lichtleitfasern zur kontinuierlich verteilten Temperatur-und Dehnungsmessung auf Basis der Frequenzbereichsanalyse (Shaker, Aachen, 2000) F.J. Harris, Proc. IEEE 66, 1 (1978) K. Krebber, Ortsauflösende Lichtleitfaser-Sensorik für die Temperatur und Dehnung unter Nutzung der stimulierten Brillouin-Streuung basierend auf der Frequenzbereichsanalyse (RUB, Bochum, 2001) A. Minardo, R. Bernini, L. Zeni, Opt. Express 22, 14 (2014) A. Minardo, R. Bernini, R. Ruiz-Lombera, J. Mirapeix, J.M. Lopez-Higuera, L. Zeni, Opt. Express 24(26), 29994–30001 (2016) M. Nikles, L. Thevenaz, P. Robert, J. Lightwave Technol. 15(10), 1842–1851 (1997) N. Nöther, Distributed Fiber Sensors in River Embankments: Advancing and Implementing the Brillouin Optical Frequency Domain Analysis (BAM, Berlin, 2010) N. Nöther, A. Wosniok, K. Krebber, Proc. SPIE 6933, 69330T-1–69330T-9 (2008a) N. Nöther, A. Wosniok, K. Krebber, Proc. SPIE 7003, 700303.1–700303.9 (2008b) S. Rashleigh, J. Lightwave Technol. 1(2), 312–331 (1983) M.A. Soto, L. Thévenaz, Opt. Express 21(25), 31347–31366 (2013) A. Wosniok, Untersuchungen zur Unterscheidung der Einflussgrößen Temperatur und Dehnung bei Anwendung der verteilten Brillouin-Sensorik in der Bauwerksüberwachung (TUB, Berlin, 2013) A. Wosniok, N. Nöther, K. Krebber, Procedia Chem. 1(1), 397–400 (2009) A. Yaniay, J.-M. Delavaux, J. Toulouse, J. Lightwave Technol. 20(8), 1425–1432 (2002)

Distributed Brillouin Sensing: Correlation-Domain Techniques

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Weiwen Zou, Xin Long, and Jianping Chen

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distributed Brillouin-Based Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brillouin Scattering in Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Principle of Brillouin-Based Distributed Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlation-Domain Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BOCDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BOCDR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Advances in Distributed Brillouin Correlation-Domain Sensing . . . . . . . . . . . . . . . . . . . . . . Effective Sensing Points Enlargement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noise Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strain and Temperature Discrimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The Brillouin based distributed sensors have great potentials in many fields such as smart materials and structers. The correlation domain technique developed in the last two decades is based on the measurement for the correlated-peaks along the optical fiber. It have attracted much attention as its unparalleled advantages in ultra high resolution and the ability to achieve dynamic measurement, which are difficult for the traditional time domain techniques. Both spontaneous Brillouin scattering and stimulated Brillouin scattering can be utilized in correlation domain techniques. Efforts have been paid to improve the performances including the effective sensing points enlargement and noise suppression.

W. Zou () · X. Long · J. Chen State Key Laboratory of Advanced Optical Communication Systems and Networks, Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 G.-D. Peng (ed.), Handbook of Optical Fibers, https://doi.org/10.1007/978-981-10-7087-7_9

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Moreover, the correlation domain techniques can be well combined with other techniques such as time domain techniques or Brillouin dynamic grating techniques.

Preface Distributed Brillouin optical fiber sensors have attracted much attention and are thought potential due to their unparalleled abilities to measure the environment (such as temperature and strain) along an optical fiber. The capacity to detect continuously along the optical fiber is superior to the pointed or multiplexed fiberoptic sensors based on fiber Bragg grating and/or inline Fabry-Perot resonator. The Brillouin scattering utilized for optical fiber sensing is a nonlinear optical process and has many advantages such as high accuracy due to the narrowband of the Brillouin gain spectrum (BGS), simultaneously sensitive to both temperature and strain, and immunity to the electro-magnetic interference. Among the Brillouinbased distributed optical sensing techniques, the correlation-domain technique utilizes continuous lightwave as pump and measures the Brillouin gain spectrum information from the correlation peak. Compared to the conventional pulse-based time-domain technique, it can achieve centimeter or even millimeter resolution and fast dynamic measurement, which is difficult for time-domain technique due to the pulsed-pump-based sensing mechanism. This chapter introduces the distributed Brillouin sensing correlation-domain techniques in optical fibers. The basic principle, working mechanism of two types of correlation-domain techniques, and methods proposed to improve the sensing ability are demonstrated, respectively.

Distributed Brillouin-Based Sensors Brillouin Scattering in Optical Fibers Brillouin scattering is a “photon-phonon” interaction as annihilation of a pump photon creates a Stokes photon and a phonon simultaneously (Agrawal 2012). The created phonon is the vibrational modes of atoms, also called a propagation density wave or an acoustic phonon/wave due to the electrostriction effect. In a silica-based optical fiber, Brillouin Stokes wave propagates dominantly backward (Ippen and Stolen 1972) although very partially forward (Shelby et al. 1985). The frequency (9–11 GHz) of Stokes photon at 1550-nm wavelength is dominantly downshifted due to Doppler shift associated with the forward movement of created acoustic phonons (Fig. 1). Brillouin scattering is basically categorized into two types: spontaneous Brillouin scattering (SpBS) and stimulated Brillouin scattering (SBS) in optical fibers. In principle, the SpBS is caused by a noise fluctuation and influences the pump wave (Ep ). The SBS occurs when the pump power for SpBS is beyond the so-called Brillouin threshold value (Pth ) or when two coherent waves with a

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Rayleigh Stokes components

Anti-stokes components

~9-11 GHz ~13 THz

Brillouin

Brillouin

Raman

Raman

f0

frequency

Fig. 1 Schematic spectrum of scattered light resulting from three scattering processes in optical fibers (After Zou et al. (2015). From Mizuno et al. (2008, 2010), Song et al. (2008, 2009b), and Elooz et al. (2014))

frequency difference equivalent to the phonon’s frequency are counter-propagated. Brillouin scattering dynamics in optical fibers are generally governed by the following coupling equations (Gaeta and Boyd 1991; Jenkins et al. 2007): 8 1 ˆ ˆ < vg ˆ ˆ :



@ C @z@ Ep D  ˛2 Ep C i 1 Es @t  1 @ @ Es D  ˛2 Es C i 1  Es   @vg @t B @z  C 2 C 2i B  D i 2 Ep Es  C @t

;

(1)

N

where Ep and Es stand for the normalized slowly varying fields of pump and Stokes (or probe) lightwaves, respectively;  denotes the acoustic (or phonon) field in terms of the material density distribution; N represents the random fluctuation or white noise in position and time (Gaeta and Boyd 1991); vg is the group light velocity in the fiber; ˛ is the fiber’s propagation loss;  B is the damping rate of the acoustic wave, which equals to the reciprocal of the phonon’s lifetime (1/ B D   D 10 ns) and is related to the acoustic linewidth ( B D  B / ) (Pine 1969); and  1 and  2 are the coupling coefficients among Ep , Es , and . If SpBS is considered, it is reasonable to assume Es is sufficiently small so that the second term of N dominates in the right side of Eq. 1. In contrast, the first term of i 2 Ep Es * dominates for SBS. Concerning the SBS power transfer between pump lightwave Pp and probe lightwave Ps under the assistance of the acoustic wave and the so-called acoustooptic effect, Eq. 1 can be rewritten as 8  < 1 @ C @ C ˛ Pp D g ./ Pp Ps v @t @z g   : 1 @  @ C ˛ Ps D Cg ./ Pp P vg @t @z

(2)

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where the sign difference between the right hands means that the pump power is reduced or depleted but the probe (Stokes) power is increased or amplified. And g(v) is called BGS, the key phraseology to represent Brillouin scattering in optical fibers. It denotes the spectral details of the light amplification from strong pump wave to weak counter-propagating probe/Stokes wave in SBS or those of the noiseinitialized scattered phonons in SpBS. There are three basic parameters of Brillouin frequency shift (BFS, B ), Brillouin gain peak value (gB0 ), and Brillouin linewidth ( B ) in the main-peak BGS. B is defined as B D

2neff Va 

(3)

where  is the light wavelength, neff is the effective refractive index of the fiber, and Va is the effective acoustic velocity of the fundamental acoustic mode. Brillouin gain peak value gB0 is determined by gB0 D

4neff 8 p12 2 3 0 cB B0

(4)

where 0 is the density of silica glass (2202 kg/m3 ) and p12 the photoelastic constant (0.271). In most silica-based fibers, the peak gain value of gB0 lies in the range of 1.53  1011 m/W (Nikles et al. 1997).  B0 in silica optical fibers with a typical value of 30–40 MHz is characteristic of SpBS. However, in the SBS process, it was theoretically proved that  B strongly depends on the pump power, which is expressed as follows (Gaeta and Boyd 1991; Yeniay et al. 2002): s B D B0

ln 2 Gs

(5)

where Gs is the single pass gain experienced by the weak probe wave from the strong pump wave.

Principle of Brillouin-Based Distributed Sensing Sensing of Measurands As a nondestructive attenuation measurement technique for optical fibers, the SBS distributed measurement could measure attenuation distribution along the fiber having no break from an interrogated optical power as a function of time, but it has much higher signal-to-noise ratio (more than 10 dB) than optical time-domain reflectometry (OTDR) due to SBS high gain. For this measurement SBS process was performed by injecting an optical pulse source and a continuous-wave (CW) light into two ends of fiber under test (FUT). When the frequency difference of the pulse

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pump and CW probe is tuned offset around B of the FUT, the CW probe power experiences Brillouin gain from the pulse light through SBS process. Furthermore it was found that this nondestructive attenuation measurement can be extended into a frequency-resolved technique because B of optical fibers has linear dependence on measurands of strain and temperature as follows (Horiguchi et al. 1989; Kurashima et al. 1990): B  B0 D A  ı" C B  ıT

(6)

where B0 is measured at room temperature (25 ı C) and in the “loose state” as a reference point, ı" is the applied strain, and ıT is the temperature change. The “loose state” means that the FUT is laid freely in order to avoid any artificial disturbances. A (or C " ) is the strain coefficient in a unit of MHz/ ", and B (or CT " ) is the temperature coefficient in a unit of MHz/ o C. Fig. 2 illustrates the characterized strain or temperature dependence in a standard single-mode fiber (SMF), where the BGS always moves toward higher B and its gain reduces or increases when ı" or ıT is increased, respectively. At 1550 nm, A D 0.04– 0.05 MHz/ " and B D 1.0–1.2 MHz/o C, which depends on the fiber’s structure and jackets. Note that Eq. 6 is the basic sensing mechanism of Brillouin-based distributed sensors. 10

(a)

9

De = 0 me De = 421 me De = 842 me De = 1263 me De = 1684 me

Gain (a.u.)

8 7 6 4

7 6 5 4

3

3

2

2

1

1 10.80

10.85

10.90

10.95

11.00

T= 8.0oC T=25.0oC T=35.0oC T=44.5oC T=57.9oC

8

5

0 10.75

(b)

9

Gain (a.u.)

10

11.05

0 10.75

10.80

10.85

Frequency (GHz) 10920 Measured data Linear fitting, 0.0403 MHz/me

BFS (MHz)

10970

(c) SMF

10930 10910

11.00

11.05

(d)

10900

10950

10.95

Measured data Least-squares fitting, 1.119 MHz/oC

10910

BFS (MHz)

10990

10.90

Frequency (GHz)

10890

SMF

10880 10870 10860

10890

10850 10870 0

400

800

1200 1600 2000 2400 2800 3200

Strain (me)

0

10

20

30

40

50

60

70

Temperature (oC)

Fig. 2 (a) Strain and (b) temperature dependences of BGS in SMF; (c) strain and (d) temperature dependences of Brillouin frequency shift B in SMF (From Song et al. (2008), Zou et al. (2009b), and Mizuno et al. (2010))

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Sensing of Location Besides the sensing of measurands, the mapping of spontaneous or stimulated Brillouin scattering process (not just nondestructive attenuation measurement (Horiguchi and Tateda 1989)) is another key issue to realize distributed optical fiber sensing (Horiguchi et al. 1990; Kurashima et al. 1993; Horiguchi et al. 1993). Two different mapping ways, as schematically illustrated in Fig. 3, were proposed. One is to repeat the localized BGS in scanned positions along the FUT; the other is to repeat the Brillouin interaction under different frequency offset. There are three different mapping or position-interrogation techniques, including time domain (Horiguchi et al. 1990; Kurashima et al. 1993; Horiguchi et al. 1993; Niklès et al. 1996; Shimizu et al. 1994; Bao et al. 1993), frequency domain (Garcus et al. 1997; Garus et al. 1996), and correlation domain (Hotate and Hasegawa 2000; Hotate and Tanaka 2002; Mizuno et al. 2008). Regarding the injection ways of optical fields, there are two opposite groups, i.e., analysis versus reflectometry. The analysis is two-end injection based on SBS; while the reflectometry is one-end injection based on SpBS. Note that there is an additional method between analysis and reflectometry, called one-end analysis (Niklès et al. 1996; Song and Hotate 2008; Zou et al. 2010). The basic principle of time-domain sensing technique is the “time-of-flight” phenomenon in FUT. For two-end or one-end analysis, named Brillouin optical time-domain analysis (BOTDA) (Horiguchi et al. 1990; Horiguchi et al. 1993; Bao et al. 1993), one of pump and probe waves is pulsed in time and the other is continuous wave (CW). Subsequently, they are successively interacted along the FUT during the time of flight of the pulsed wave. In contrast, for oneend reflectometry, called Brillouin optical time-domain reflectometry (BOTDR) (Kurashima et al. 1993; Shimizu et al. 1994), the pump wave is pulsed in time, and the SpBS Stokes wave is reflected along the FUT during the pump’s time of flight. The spatial resolution (ZTD ) of time-domain distributed sensing is physically determined by the pulse width ( ) (Horiguchi and Tateda 1989): ZTD D

c 2n

(7)

Frequency offset

Fig. 3 Schematic of sensing of location or mapping of BGS (After Zou et al. (2015). From Song et al. (2009b))

45 Distributed Brillouin Sensing: Correlation-Domain Techniques

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where n is the group velocity of the pulse. The BGS mapping is realized by repeating the above measurement when the spectrum of the reflected Stokes in BOTDR is processed or the optical frequency offset between the pump and probe in BOTDA is tuned around the BFS B . There are two kinds of correlation-domain sensing techniques, nominated Brillouin optical correlation-domain analysis (BOCDA) (Hotate and Hasegawa 2000; Hotate and Tanaka 2002) and Brillouin optical correlation-domain reflectometry (BOCDR) (Mizuno et al. 2008; Mizuno et al. 2010a). In experiment, the BOCDR and BOCDA can be executed by substituting a distributed feedback laser diode (DFB-LD) driven by a function generator (such as in a sinusoidal function) for the light source. The optical frequency offset between pump and probe or between scattered Stokes and optical oscillator changes with time as well as position, deviating from the preset constant frequency offset around the BFS B . The spatial resolution of BOCDA and BOCDR are both determined by (Hotate and Hasegawa 2000) ZCD D

c B  2nf m f

(8)

where fm is the modulation frequency of the sinusoidal function, f the modulation depth, and  B the Brillouin linewidth defined in Eq. 5.The maximum measurement length (or sensing range, LCD ) is decided by the distance between two neighboring correlation peaks (Hotate and Hasegawa 2000): LCD D

c 2nf m

(9)

Because of the difference of the physical pictures between time domain and correlation domain, their sensing performance is different. For example, the spatial resolution of BOTDA/BOTDR was typically limited to be 1 m by the lifetime of acoustic phonons (10 ns) and the nature of intrinsic Brillouin linewidth. However, BOCDA/BOCDR is of CW nature free from this limitation, and their spatial resolution can be cm-order (Hotate and Tanaka 2002; Mizuno et al. 2009a) or even mm-order (Song et al. 2006a).

Correlation-Domain Technique BOCDA In an original BOCDA technique, the frequencies of the pump and the counterpropagating probe lightwaves are both sinusoidally modulated. As shown in Fig. 4, at specific positions along the optical fiber, these two lightwaves undergo synchronous frequency modulation, and the beat frequency maintained a constant value (Hotate and Ong 2002). Furthermore, if this time-invariant frequency is

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a

(n-1)th

nth

(a)

(n+1)th

Correlation (b)

fiber under test pump

probe

probe

νB

pump (viewed from probe)

gain

~DnB

nB Synchronous modulation

b

n

nB

Asynchronous modulation Fig. 4 Illustration of the correlation of the sinusoidally modulated pump and probe lightwaves. The Brillouin gain spectra are different between (a) correlation-peak positions and (b) noncorrelation positions (After Hotate and Ong (2002). From Mizuno et al. (2008), Zou et al. (2009b), and Song et al. (2009b))

close to the local BFS (11 GHz for a standard single-mode fiber), the acoustic wave will be strongly and continuously stimulated. For these positions, they are called “correlation-peak points” or “correlation-peak positions.” The Brillouin gain spectrum for correlation-peak position possesses narrow bandwidth of Lorentz shape (see Fig. 4a). As for the non-correlation positions, the pump and probe lightwaves experience asynchronous frequency modulations. From the view of the pump lightwave, the frequency of the probe lightwave swings broadly around. The beat frequency needed to maintain the acoustic wave is not satisfied, resulting a flat weak Brillouin gain spectrum (see Fig. 4b). By sweeping the frequency difference between two lightwaves detecting the probe lightwave, the measured gain spectrum is the integration of the BGS along the optical fiber, which mainly comes from the correlation-peak positions and the non-correlation positions which contribute few to the final result. If only one peak

45 Distributed Brillouin Sensing: Correlation-Domain Techniques

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exists along the optical fiber, the BFS of the measured BGS is a function of the measurands from the correlation-peak position. Considering a sinusoidal frequency modulation is modulated on both pump and probe lightwaves, the modulation frequency is fm , the frequency deviation is f, and the mean frequency difference between two lightwaves is v. As the two lightwaves are counter-propagating along an optical fiber, the temporal delay  between them depends on the longitudinal position as D

2x vg

(10)

where x represents the distance from correlation peak (it could be any one). Then the beat frequency between these two lightwaves can be written as fbeat .t / D v C f fsin .2fm t /  sin Œ2fm .t  / g D v C 2f cos .f i  m / sinŒfhm .2t / sin 2fm t  x D v C 2f cos 2fm x vg vg

(11)

The beat frequency fbeat (t) is still a frequency modulated signal with modulation frequency fm and frequency deviation 2f cos(2 fm x/vg ). To ensure the correlation peak, the beat frequency between pump and probe lightwaves should be restricted in an acceptable bandwidth to generate a strong and stable acoustic wave, which can be considered as the FWHM of the intrinsic BGS vB . If condition x < < vg /fm is satisfied, this bandwidth restriction can be expressed as 2fm f

x  vB vg

(12)

It can be considered that the final measured gain spectrum is mainly contributed by the positions that satisfy Eq. 12. Thus the spatial resolution of the correlationdomain technique is given by z D

vg vB 2fm f

(13)

which is the same as Eq. 8. Thus the spatial resolution depends on both the modulation frequency fm and the frequency deviation f, which can be very small by carefully selecting these two values. For instance, if fm D 50 MHz and f D 1 GHz and the vB is usually around 30 MHz, the calculated sensing resolution according to Eq. 13 is around 2 cm, which is ten times better than the basic pump-pulsed-based time-domain techniques. On the other hand, the 0th correlation peak is always located at the center of the optical fiber, i.e., the zero path difference point for two lightwaves. While the other non-0th correlation-peaks’ position are periodically distributed along the fiber and are dependent on the parameter of the frequency modulation. According to Eq. 11,

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the distance dm between the adjacent correlation peaks will be given by dm D

vg 2fm

(14)

By scanning this modulation frequency, the positions of the non-0th correlation peaks can be changed continuously along the optical fiber, and the distributed information can be obtained. It should be noted that, as mentioned above, the correlation-domain sensing needs to ensure that only one correlation peak contributes to the final gain spectrum’s measurement. This means that the sensing range for this technique cannot exceed dm ; otherwise two or more correlation peaks occur along the optical fiber and the measurement will be confused. According to the definition of Eq. 14, the sensing range of a basic BOCDA technique is dependent on the modulation frequency, usually several to tens of meters in applications, which is very short compared with the time-domain techniques. Figure 5 gives an illustration explaining the main configuration of the basic BOCDA technique (Yamauchi and Hotate 2004). Sinusoidal frequency modulation is implemented by direct modulation onto the laser, and the generated lightwave is divided into two arms by a coupler. The loop configuration ensures the counterpropagation between two arms, and the frequency shifter is used to tune the frequency different. Thus the beat frequency between pump and probe lightwaves is close to the BFS of the FUT. Tunable delay is utilized to shift the correlation peak to be measured. Isolator and circulator limit the optical fiber that the stimulated Brillouin scattering occurs, i.e., the FUT, to ensure one and only one correlation peak exists. The amplified probe lightwave is finally received by the photodetector, and the gain spectrum is obtained by simply scanning the beat frequency of the frequency shifter. Unlike the time-domain techniques, the measurements of correlation-domain system are continuously processed due to the CW of both two lightwaves and independent on the optical fiber length. For the sensing at one particular position, the measurement period is mainly limited by sweeping time of the frequency shifter in

Fig. 5 The basic configuration of the BOCDA technique (After Yamauchi and Hotate (2004). From Mizuno et al. (2008) and Zou et al. (2009a))

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order to obtain the BGS, which can be done in milliseconds or even faster. Moreover, thanks to the CW design, the acoustic wave is well maintained and measured Brillouin gain is strong enough. Thus the averaging work is not necessary for a correlation-domain technique. In this way, the Brillouin-based correlation-domain sensors have the ability to implement the dynamic measurement.

BOCDR Like other Brillouin-based distributed sensing techniques, such as BOTDA and BOTDR, the correlation-domain techniques also include two types depending on whether the spontaneous or stimulated Brillouin scattering is measured. The following introduced BOCDR utilizes the spontaneous Brillouin scattering by the pump lightwave to implement the distributed sensing. The most advantage for BOCDR compared with BOCDA technique is the ability to work with one-end access to the fiber. Figure 6 is a schematic of a basic BOCDR system (Mizuno et al. 2008). The lightwave generated by the light source is divided into pump and reference lightwaves by an optical coupler. The pump lightwave is launched into the FUT from one end, and the reflected Stokes lightwave by the SpBS is directed into a heterodyne receiver composed of two balanced photodiodes (PDs), while the reference lightwave is used as an optical local oscillator to be processed with the Stokes lightwave. The electrical beat signal of the two lightwaves is monitored by an electrical spectrum analyzer (ESA). Since there is a frequency difference of about 11 GHz between the Stokes light and the reference light, this configuration is called self-heterodyne scheme. Recently, an ultrahigh-speed configuration of BOCDR with a strain sampling rate of up to 100 kHz has been developed (Mizuno et al. 2016).

Fig. 6 Conceptual schematic of BOCDR technique (After Mizuno et al. (2008). From Elooz et al. (2014) and Song et al. (2009b))

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Unlike the BOCDA system, the measured BGS of the correlation-peak position is directly obtained without frequency sweeping. In order to resolve the position in the FUT, the optical frequency of the light beam from the laser is directly modulated in a sinusoidal wave by modulating the injection current to the laser. From the viewpoint of time averaging, the correlation (or coherence) function is synthesized into a series of periodical peaks, whose period is inversely proportional to the frequency of the sinusoidal modulation fm , which is similar to the case in BOCDA scheme. By controlling fm to leave only one correlation peak within the range of the FUT, the Brillouin scattering generated at the position correspondent to the peak has high correlation with the reference lightwave and then gives high heterodyne output. The peak frequency observed in the ESA gives the BFS caused at the position. By sweeping fm , the correlation peak is scanned along the FUT to obtain the distribution of BGS or BFS. Figure 7 shows the measured distributed BGS along the FUT by a fabricated basic BOCDR system as an example (Mizuno et al. 2008). In the system, a 100m-long SMF is utilized as the FUT. The modulation frequency fm is set as 457.4– 458.4 kHz and the frequency deviation f is 5.4 GHz. A 0.2-% strain is applied onto a 50-cm-long section along the FUT, while the system’s spatial resolution should be 40 cm according to Eq. 13, which is also suitable for BOCDR system. The overall sampling rate of the BGS measurement for a single position was 50 Hz, which is much higher than that of the time-domain techniques (typical measurement time several minutes).As it is shown in the distribution of the BGSs, the BFSs are shifted to around 11 GHz away from the original 10.8 GHz within a small section. The corresponding BFS along the FUT is plotted in Fig. 8. A 50-cm section is found with BFS shifted, which agrees with the experimental setup. The measured strain can be determined by the frequency shift of the BFS and the strain coefficient of the

Fig. 7 Distribution of the BGS along the FUT (After Mizuno et al. (2008, 2010))

Fig. 8 Distribution of the BFS (peak of the BGS) along the FUT (After Mizuno et al. (2008))

Brillouin frequency shift [GHz]

45 Distributed Brillouin Sensing: Correlation-Domain Techniques

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11.02

10.98 50 cm 10.94

10.9

10.86 96

96.5 97 Position [m]

97.5

SMF. The change of the BFS was about 100 MHz, which is in good agreement with the applied strain of 0.2%.

Advances in Distributed Brillouin Correlation-Domain Sensing Effective Sensing Points Enlargement Despite the high spatial resolution, the basic BOCDA/BOCDR systems can only measure the optical fiber no more than hundreds of meters. The measurement range of the Brillouin-based distributed correlation-domain technique is limited by the “one-peak” condition and given as defined as Eq. 14. According to that, if lower modulation frequency fm is applied, longer detection range we get. It seems easy to achieve a long sensing range as applying low fm isn’t difficult. However, noted that fm also affects the spatial resolution as Eq. 13, lower fm results in a worse sensing resolution. It is meaningless to give up the original incomparable resolution just in order to enlarge the measurement range. Thus this modulation parameter introduces a trade-off problem on the range-resolution issue. Through dividing the measurement range by the spatial resolution, factor fm is eliminated as N D

dm f D z vB

(15)

where N is called the effective sensing points. Meanwhile, the original proposed correlation-domain scheme measures the local BGS along the optical fiber, and all information from the correlation-peak positions will be integrated. A better way to enlarge the measurement range is to allow many correlation peaks along the fiber under test but only detect the gain spectrum from single one of them. Many efforts have been made utilizing this basic principle to

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enlarge the sensing range, such as double modulations (Mizuno et al. 2010b; Zou et al. 2011), double lock-in amplifiers (Song and Hotate 2006), temporal gating (Kannou et al. 2003; Yamashita et al. 2012; Mizuno et al. 2009b), sensing fiber with different types (Jeong et al. 2011, 2012), and combination with the time-domain technique (Elooz et al. 2014; London et al. 2016; Denisov et al. 2016). In the following several schemes will be introduced.

Double Modulations The basic idea of using double-modulation scheme is to generate two-pattern correlation-peaks distribution (Mizuno et al. 2010b). In this way, the spatial resolution and measurement rang are decided by different modulation parameters. Figure 9 shows the experimental setup of a BOCDR system based on the doublemodulation scheme. Two frequency modulations are applied on the pump and reference lightwaves, respectively. The modulation frequencies are f0 (Cf" ) and mf0 , where f0 is a fundamental frequency, m is an integer, and f" (0.5 kHz) is needed to avoid beating between the two frequencies, which causes large fluctuations of BGS. The amplitude of the frequency modulation at f0 , denoted as f1 , is set to be several hundreds of MHz (difficult to measure accurately due to the frequency characteristics of the laser circuit). The amplitude at mf0 , denoted as fm , is about 5.4 GHz (a little lower than a half of BFS in silica fibers). Figure 10 shows the correlation-peaks distributions caused by the above two frequency modulations. For f0 (Cf" ) case, the correlation peaks have a larger distance from each other, while in mf0 case, the correlation peaks are of narrow linewidth. As the final measured result

Fig. 9 Experimental setup of the BOCDR technique with double-modulation scheme. DAQ, data acquisition; DC, direct current; EDFA, erbium-doped fiber amplifier; ESA, electrical spectrum analyzer; FUT, fiber under test; GPIB, general-purpose interface bus; PC, polarization controller; PD, photodetector; PSCR, polarization scrambler (After Mizuno et al. (2010b))

45 Distributed Brillouin Sensing: Correlation-Domain Techniques

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Fig. 10 Operating principle of double-modulation scheme (After Mizuno et al. (2010b))

Power [µV]

0.5 0.4 0.3 0.2 0.1 0 989 989.5

991.5 10.84 10.88 10.92 10.96 Brillouin frequency shift [GHz]

[m on siti

991

Po

990.5

]

990

b Brillouin frequency shift [GHz]

a

10.96 10.94 10.92 10.9

40 cm

10.88 10.86 10.84 10.82 988.5 989 989.5 990 990.5 991 991.5 992 Position [m]

Fig. 11 Measured distributions of (a) BGS and (b) BFS with the double-modulation scheme when the noise-floor compensation technique was employed (N D 5690) (After Mizuno et al. (2010b))

comes from the correlation-peaks distribution combined by these two cases, high spatial resolution and long detection range can be simultaneously achieved. The N is increased by m times according to the theoretical analysis. Figure 11 gives the experimental results obtained by the double-modulation scheme with f0 and 10f0 (f10 D 5.4 GHz) using a noise-floor compensation technique (Mizuno et al. 2009c). A strain of 0.15% was applied to a 40-cm section (990.0–990.4 m). With f0 swept from 68.362 kHz to 68.474 kHz, the spatial resolution and the measurement range were 26.5 cm and 1.51 km, respectively, corresponding to N of 5690. The measured results are shown in Figs. 11a, b. The strain-applied 40-cm section was successfully detected. The measurement accuracy was about C/ 15 MHz, corresponding to about C/ 260 ".

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Double Lock-in Amplifiers According to Eq. 15, a direct way to increase the effective points N is to apply the frequency modulation with larger f (Song and Hotate 2006). However, it has been practically limited to around 5 GHz, less than the half of the vB of the fiber, over which the spectrum of the pump and the probe waves starts to overlap. If this condition is not satisfied, the backward reflection of the pump lightwave acts as a terrible noise for probe lightwave due to the indistinguishable spectrum. Besides, in case of using a typical EOM to generate a stable frequency difference between the pump and the probe waves, the system additionally suffers the reduction of signal if f > vB , where the overlap of the spectrum starts between the two sidebands from the EOM. Since they give opposite effects (gain and loss) on the BGS measurement, this condition results in a strong decrease of probe signal by the counterbalancing of Brillouin gain and loss. In order to avoid the unwanted overlaps, the reflection of the pump lightwave must be filtered out from the probe lightwave. A usual way is to utilize an optical filter before the detection (see Fig. 12a). However, this method cannot distinguish the reflection component whose frequency is within the probe’s spectrum. Figure 12b demonstrates a schematic of the double lock-in detection, while the previous single lock-in configuration is shown in Fig. 12a. In addition to the lock-in detection with the pump wave, another lock-in detection is applied to the probe wave at different lock-in frequency, which removes the backward reflection of pump waves from the probe signal without using any optical filter. Therefore, the double lock-in detection works effectively even in the situation when no optical filtering is available due to a large modulation (f > vB /2) of the laser source.

Fig. 12 Measurement schemes for the BOCDA system using (a) single lock-in amplifier and (b) double lock-in amplifiers. Note that the optical filter in (a) is not available if the laser modulation amplitude is larger than vB /2 (After Song and Hotate (2006))

45 Distributed Brillouin Sensing: Correlation-Domain Techniques

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Fig. 13 Experimental setup for the BOCDA system using double lock-in amplifiers and an SSB modulator. PSW, polarization switch (After Song and Hotate (2006))

The experimental setup is depicted in Fig. 13. Based on the original BOCDA scheme with single lock-in amplifier, the probe branch is also chopped using an additional EOM, and the second lock-in amplifier is used to filter out the reflection of the pump lightwave. The modulation frequency fm of the laser was 310–320 kHz depending on the position of the correlation peak in the fiber, which corresponds to the measurement range of more than 320 m according to Eq. 14. The amplitude of the frequency modulation was 15.5 GHz which was larger than vB of conventional fibers (10.8 GHz), and the spatial resolution of the measurement was calculated to be about 20 cm from Eq. 13. The N of sensing points is about 1500 according to Eq. 15. Experimental results are shown in Fig. 14 with double lock-in amplifiers comparing with the results from single lock-in amplifier scheme. BGSs measured for two positions within SMF (upper) and DSF (lower) are demonstrated in Fig. 14a. It is clear that for the single lock-in detection case, the measured BGS is seriously affected by the noise with a large background dc component, which can be attributed to the backward reflection of the pump lightwave. Especially in the SMF, the wanted BGS is buried in the noise and the BFS cannot be determined, while the double lock-in detection scheme can effectively suppress the reflection noise and the BFS is more easily valued. Figure 14b shows the plot of the Brillouin peak frequencies around one of the DSF sections for the single lock-in (dashed line) and double lock-in (solid line) configurations. A large fluctuation of peak frequencies is observed in the case of single lock-in detection due to the noises. These results confirm the effectiveness of the double lock-in detection in a long range measurement.

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Fig. 14 Comparison of the data in the case of the single and the double lock-in detections. (a) The BGS graphs corresponding to the SMF (upper) and the DSF (lower) sections of the FUT. (b) Measured Brillouin peak frequencies around the DSF section of the FUT in the single (dashed line) and the double (solid line) lock-in systems. (After Song and Hotate (2006))

Combination with Time-Domain Technique The effective points N defined in Eq. 15 is obtained from the definition of Eqs. 13 and 14, which give the original spatial resolution and measurement range, respectively (Elooz et al. 2014). When fm is large, the distance between the correlation peaks is small and the detection range is limited. Otherwise the measured BGS will be the integration of several BGSs of correlation peaks distributed along the FUT. So if there is a way to distinguish the BGS from each correlation peak, the measurement range of the system is no longer limited by Eq. 14, and thus a large N can be achieved. Luckily, the time-domain techniques happen to have the ability to translate the distribution information along the FUT, and thus the combination of these two techniques is raised. Denote the complex envelopes of the pump and probe lightwaves as Ap (z,t) and As (z,t), respectively, where z denotes position along a fiber of length L and t represents time. The pump wave enters the fiber at z D 0 and propagates in the positive z direction, whereas the signal wave propagates from z D L in the negative z direction. In the proposed scheme, the signal envelope at its point of entry into the fiber is modulated by a phase sequence cn with a symbol duration T that is much shorter than the acoustic lifetime: As .z D L; t / D as

X n

Ap .z D 0; t / D ap rect



t  nT cn rect T

  As0 .t /

  X  t t  nT  Ap0 .t / cn rect  T n

(16)

(17)

45 Distributed Brillouin Sensing: Correlation-Domain Techniques

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where ap and as are the constant magnitudes of the pump and probe lightwaves and  is the duration of the pump amplitude pulse. The magnitude of the acoustic field at a given location and time is given by Z Q .z; t / D jg 1

   z z As  a  exp ŒA .t  a/ Ap a   .z/ da vg vg 0 (18) t

where g1 is a parameter which depends on the electrostrictive coefficient, the speed of sound, and the density of the fiber, vg is the group velocity of light in the fiber, and the position-dependent temporal offset z is defined as z D (2z–L)/vg . The distribution and variation of the acoustic wave along the FUT are important for the understanding and analysis of a Brillouin-based system. Equation 18 can be solved with simple waveforms given for pump and probe lightwaves, or it can be just numerically integrated, subject to the boundary conditions of Eq. 16 and Eq. 17. Under the conditions (including the modulation of the pump and probe lightwaves) described above, the magnitude of the acoustic wave over a 6-m-long fiber is illustrated in Fig. 15. The BFSs along the FUT are the same, and the frequency difference between the pump and probe is chosen to match this value. A perfect Golomb code (N D 127, T D 200 ps) is used in the phase modulation of both pump and probe, and a 26 ns-long amplitude pulse was superimposed on the

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Fig. 15 Simulated magnitude of the acoustic wave density fluctuations (in normalized units), as a function of position and time along a 6-m-long fiber section. Both pump and signal waves are comodulated by a perfect Golomb phase code that is 127 bits long, with symbol duration of 200 ps. The pump wave was further modulated by a single amplitude pulse of 26 ns duration. The acoustic field, and hence the SBS interaction between pump and signal, is confined to discrete and periodic narrow correlation peaks. The peaks are built up sequentially one after another with no temporal overlap (After Elooz et al. (2014))

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Fig. 16 Measured Brillouin gain map as a function of frequency offset between pump and signal and position along a 400-m-long fiber under test. The fiber consisted of two sections, each approximately 200 m long, with Brillouin shifts at room temperature of approximately 10.84 GHz and 10.90 GHz, respectively. A 5-cm-long hot spot was located toward the output end of the pump wave. The map was reconstructed using only 127 scans per frequency offset, according to the combined BOTDA/BOCDA method. The complete map is shown on panel (a), and a zoom-in on the hot spot region is shown on panel (b) (After Elooz et al. (2014))

phase-modulated pump wave. As it is shown, three correlation peaks are generated along the fiber with same distance, which agrees with the basic theory of a BOCDA scheme. However, these correlation peaks do not last for a long period; instead they only exist for no more than 20 ns. It should be noted that the “moving” direction of the probe lightwave is from the right-top to the left-button in a z-t distribution map. The Brillouin gain comes from the gathering of the interaction between the pump and acoustic wave all along the probe lightwave’s path. In this way, the detected probe gain can distinguish the information from different correlation peaks due to their short existing duration, resulting in the allowance for the multiple correlation peaks in a correlation-domain technique. The experimental measured BGS distribution along the FUT is given in Fig. 16. Measurement is implemented over a 400-m-long fiber with a 5-cm-long hot spot section. The spatial resolution is 2 cm due to the theory, and the hot spot section is clearly detected (see Fig. 16b), which means that the effective point N reaches 20,000.

Noise Suppression As it is discussed in section “Principle of Brillouin-Based Distributed Sensing,” the Brillouin-based distributed sensors include two matters: sensing of the location and

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sensing of the measurands. The effective point N associated with the measurement range and spatial resolution is the key performance on the location sensing, answering the question that how long and how precise we can measure the distributed information along the FUT. The other problem is that how accurate and how large we can determine the environment parameters (i.e., BFS) which change at specific position along the fiber. In usually, the BFS is determined by the frequency with biggest Brillouin gain in a BGS. However, it is difficult to achieve a high accuracy, and even sometimes the BFS cannot be determined from the BGS due to lots of background noises such as the reflection of the pump lightwave. A simple way to suppress the background noise in Brillouin-based distributed sensing applications is the use of Brillouin gain and loss effects (Zou et al. 2012). In this modified scheme, a dual-parallel Mach–Zehnder modulator is sufficient to generate both the Brillouin gain and loss effect and the SNR is improved by 3 dB. Besides, efforts have been paid to focus on the noise suppression in the Brillouin-based correlation-domain sensors. In the following, we will introduce the suppression of these noises to enhance the sensing performance based on the intensity modulation. As it is illustrated in Fig. 17a, the measured signal in a basic BOCDA system is the integration of the local BGS distributed along the optical fiber. As the acoustic wave is only maintained within the correlation peaks, the detected gain spectrum

Correlation peak (sensing position)

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=

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0 Dn[MHz]

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Fig. 17 (a) Schematic of a Brillouin optical correlation-domain analysis (BOCDA) system. Measured Brillouin gain spectrum (BGS) is the sum of local BGS’s (LBGS);  , frequency offset between pump and probe waves. (b) Variation of the BGS in response to the applied strain to the sensing (correlation peak) position. Note that the maximum measurable strain (dashed line) is limited by the peak of the background noise and that the measurable strain limit is decreased in longer measurement range (lower) than the shorter case (upper).  is the relative frequency offset with the initial value set to zero (After Song et al. (2006b))

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is considered mainly contributed by the BGS at these positions. However, when the measurement range is enlarged, the BGSs of the non-correlation-peak positions tend to stack higher background noise for the final measured signal, which restricts the maximum measurable strain and the measurement range of the BOCDA system by leading to the failure of sensing over certain limits (see Fig. 17b). For the purpose of suppression and the modification of the background noise of the BGS, intensity modulation (IM) can be used to modify the optical spectra of pump and probe lightwaves. Three different intensity modulation waveforms are used to generate different optical power spectra as shown in Fig. 18a. Each spectral shape can be characterized by different power distribution between the side and the center of the initial spectrum (No IM) of the sinusoidal frequency modulation. The calculated waveform was applied to the IM using an arbitrary function generator which was synchronized to the frequency modulation as depicted in Fig. 18b. The other optical spectra (IM 2, IM 3) were produced by manipulating the offset and the amplitude of the modulation waveform of the transmittance used for the IM 1. Figure 19 shows the measured BGS’s in different modulation schemes. As is clearly seen in the case of no intensity modulation (No IM), the real signal from the DSF section is lower than the noise peak, so cannot be detected properly in this condition. When the IMs are applied, strong suppression of the noise peak is observed on the DSF sections compared to the signal amplitude in all cases (IM 13). At the same time, a large dip is observed at the center of the BGS in the cases of IM 1 and IM 2, which might be the effect of over-suppression. This feature gives a problem in the peak detection of normal position by “absorbing” the signal as shown in the BGSs of their SMF sections. Experiments have been carried out to verify the scheme. Distributed measurements with and without the intensity modulation by 10-cm step on the FUT using

b Optical frequency [THz]

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Fig. 18 (a) Power spectra measured by an optical spectrum analyzer with intensity modulation Schemes (IM 13) applied in addition to the initial frequency modulation (No IM). (b) Time waveforms showing the synchronization between the frequency modulation of the LD (black) and the transmittance of the intensity modulator (red) applied to generate a flat-top spectrum (IM 1) shown in (a). Note that the other waveforms (IM 23) are synchronized in the same way (After Song et al. (2006b))

45 Distributed Brillouin Sensing: Correlation-Domain Techniques

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Fig. 19 Brillouin gain spectra measured on the DSF and the SMF sections using several intensity modulation schemes shown in Fig. 18 (After Song et al. (2006b))

the same experimental parameters are performed. The distributed BFS along the FUT is given in Fig. 20. As it can be seen, the BFS change within the small section of the DSF is detected by measurement using IM 3.

Strain and Temperature Discrimination As explained in section “Sensing of Measurands,” the sensing of the measurands are all dependent on the measurement of the localized BFS. This common mechanism results in a heavy trouble for any Brillouin-based sensors in discriminating the response to strain from that to temperature by using a single piece of fiber (Song et al. 2008). In current practices, two fibers are used to discriminate the strain and the temperature: the first one is embedded or bonded at the target material/structure to feel the total effects of strain and temperature, while the second fiber is placed beside the first one and kept in loose condition so that it feels the effect of temperature only; then the strain and the temperature can be calculated by mathematics. In the following, a new method based on the Brillouin dynamic gratings (BDG) for

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10.8 No IM IM 3

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Fig. 20 Result of distributed measurement on the fiber under test near the DSF sections. The DSF sections are properly detected only in the optimum intensity modulation (IM 3). The measurement inaccuracy of the  B at each position was about C/3 MHz (After Song et al. (2006b))

complete discrimination of strain and temperature by use of only one piece of Panda-type polarization-maintaining fiber (PMF) is demonstrated.

Principle of the BDG As given in Eq. 3, the BGS is determined by the effective refractive index in an optical fiber. While in a PMF (or any medium with birefringence), optical waves with two principal polarizations (i.e., x and y polarization) experience different vB owing to their different refractive indexes. Considering that the acoustic wave generated by SBS is a longitudinal one that is free of the transversal polarization, an interesting condition can be reached that the x- and the y-polarized optical waves in a PMF show the same vB at different wavelengths. When the dispersion of the acoustic wave is ignored, the condition is expressed by following equations: nx vx D ny vy

(19)

where nx,y and vx,y are the refractive indexes and the optical frequencies in x and y polarizations, respectively. Since the SBS-induced acoustic waves can be viewed as moving gratings for the reflection of the pump wave without polarization dependence, it is expected that

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acoustic waves generated by SBS between the x-polarized pump and Stokes waves at the optical frequency vx will show strong reflectance to the y-polarized pump wave at the frequency of vy . Considering that the intensity and the wavelength of the acoustic waves are easily tuned by controlling the x-polarized “writing” beams, one may expect the SBS in a PMF to play a role of a tunable dynamic grating. For the writing of the dynamic grating, a 1550-nm laser diode was used as a light source, and the output power was divided by a 50/50 coupler. A SSBM and a microwave synthesizer were used to generate the Stokes wave (pump2) of the writing beams, and the output was amplified and polarized by an EDFA and an x polarizer. The Brillouin pump wave (pump1) of the writing beams was prepared by amplifying the original wave with the same polarization as that of pump2. Pump1 and the pump2 were launched into a PMF in opposite direction to each other through polarization beam combiners (PBC1, PBC2). For a reading beam (probe), a tunable laser with an operating wavelength near 1550 nm was used as a light source after being polarized in the y-axis. The output was launched into the PMF in the direction of the pump1 through a polarization-maintaining circulator and PBC1. The transmitted power of the probe was measured using a power meter, and the backreflected spectrum was monitored using an OSA through a y polarizer (see Fig. 21). For the detection of the dynamic grating, the frequency of the probe was tuned at the higher frequency region while monitoring the spectrum with the OSA, and the result is shown in Fig. 22. When v (the frequency difference between pump1 and the probe) was 72.6 GHz, a large reflection of the probe was observed (black curve) as a result of the dynamic grating at the frequency detuned from the probe by the same amount as that between pump1 and pump2. When one of the pumps (pump1) was turned off, the dynamic grating disappeared as depicted by the gray curve although the probe was still propagated as confirmed by the Rayleigh scattering seen at the probe frequency. In both cases, the x-polarized pumps were

Fig. 21 Experimental setup (After Song et al. (2008))

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Fig. 22 Optical spectra monitored by an OSA in the generation of dynamic rating. The gray curve corresponds to the case that one (pump1) of the writing beams is turned off, and the black curve with both writing beams turned on (After Song et al. (2008)))

observed in spite of the use of the y polarizer in front of the OSA, which originated from the finite extinction ratio (20 dB) of the polarizing components. The small peaks near pump1 correspond to the first- and second-order anti-Stokes waves that were suppressed in the SSBM used for the generation of pump2. The principle of the completely discrimination of the strain and temperature is that the coefficients of the BFS changed by the strain and temperature are strongly independent. Figure 23 shows the experimental results as the evidence. It demonstrates that for the strain variation, the BFS and the frequency deviation v behave similarly, while for the temperature change, they react in opposite direction. All the dependences show excellent linearity; thus by linear fitting, we get the strain coefficient and the temperature coefficient of C " D C0.03938 MHz/ " and C T D C1.0580 MHz/o C for the BFS  B and Cf" D C0.8995 MHz/ " and CfT D 55.8134 MHz/o C for the frequency deviation fyx , respectively.

Combination with BDG Technique In the following, we show that the dynamic grating can be localized in an arbitrary position along the PMF by using a correlation-based CW technique (Zou et al. 2009b). A distributed measurement of the dynamic grating spectrum (DGS) is demonstrated with 1.2-m spatial resolution and 110-m measurement range. In experiment, temperature-induced changes in both the BFS of BGS and the frequency deviation v of DGS are measured in heated segments cascaded along a 110-m PMF. Figure 24 shows the experimental setup for distributed generation and detection of the DGS in a 110-m-long PMF. The output from a 1549 nm distributed feedback laser diode (DFB-LD1) is equally divided into probe and pump beams. The probe beam is prepared by downshifting its frequency using a SSBM and a microwave synthesizer. It is amplified via an EDFA and launched into the PMF after passing through an x polarizer and a polarization-maintaining isolator (PM-ISO). The pump beam is chopped by an EOM and amplified by a high-power EDFA, which is

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Fig. 23 Measured strain and temperature coefficients. (a) Strain dependence. (b) Temperature dependence. Circles denote the experimental results for Brillouin frequency shift ( B) in left vertical axes, and triangles correspond to the birefringence-determined frequency deviation (v) in right vertical axes, respectively (After Zou et al. (2009a))

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Fig. 24 (Color online) Experimental configuration for distributed generating and measuring the BGS and DGS in a PMF. The abbreviations are explained in the text body. Inset A depicts the optical power spectra of two modulated DFB-LDs both with 2f D 18 GHz, where the gray curve has 6 GHz offset from the black curve corresponding to a reduced dc injection current to DFB-LD2 (After Zou et al. (2009b))

Fig. 25 (Color online) Distributed measurement results. (a) Prepared PMF sample. (b) Examples of 3D distribution of measured BGS and DGS from 2 m to 9 m when the heater is turned on. (c) Summarization of detected BFS B (upper) or frequency deviation fyx (lower) and their temperature-induced changes near the heated segments. Solid-circle line (open-triangle line) corresponds to the heater being turned off (on); solid-star line indicates the temperature-induced changes between the heater’s on and off states (After Zou et al. (2009b))

45 Distributed Brillouin Sensing: Correlation-Domain Techniques

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launched into the other end of the PMF after passing through an x polarizer, a PM circulator (PM-CIR), and a PM beam splitter–combiner (PBS–C). Used as the readout beam, the output from the second laser diode (DFB-LD2) is launched into the PMF after passing through a y polarizer and the PBS–C to readout the SBSgenerated DGS. The frequency of the readout beam is deviated by v from the pump beam by tuning the DFB-LD2’s DC injection current. The distributed measurement ability is verified as demonstrated in Fig. 25. Four heated segments cascaded along the PMF sample are prepared (see Fig. 25a). For the distributed measurement, the modulation frequency fm is scanned from 860 kHz to 930 kHz with a step of 100 Hz corresponding to 16 cm; the microwave frequency to SSBM and the y-polarized carrier frequency are ramp swept for the characterization of BGS and DGS, respectively. As an example, the detected distribution of BGS and DGS from 2 m to 9 m is illustrated in Fig. 25b. The initial DGS is not uniform along the fiber owing to the irregularity of the local birefringence introduced during the fiber fabrication. Figure 25c summarizes the measured vB and v along the fiber sample when the heater (a hot plate) with T D 15 ı C is turned off (solid-circle line) or turned on (open-triangle line); their differences between the heater’s on–off states are also shown (solid-star line). The opposite responses of the vB and v to T can be clearly observed.

Conclusion We have introduced the principle of the Brillouin-based correlation-domain distributed sensing techniques. The basic ideas for BOCDA/BOCDR are to establish the correlation peaks with strong and stable acoustic wave and then to measure the gain spectrum from these correlation-peak positions. High spatial resolution and the ability to achieve dynamic measurement are the main advantages for correlationdomain techniques compared with the traditional time-domain techniques. Works to enlarge the measurement range and to suppress the background noise have been demonstrated.

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Part X Optical Fiber Sensors for Industrial Applications

Optical Fiber Sensors for Remote Condition Monitoring of Industrial Structures

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Tong Sun OBE, M. Fabian, Y. Chen, M. Vidakovic, S. Javdani, K. T. V. Grattan, J. Carlton, C. Gerada, and L. Brun

Contents Fiber Bragg Grating (FBG)-Based Sensing Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FBG-Based Sensors for Monitoring Full-Scale Marine Propellers . . . . . . . . . . . . . . . . . . . . Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-Sensing Electric Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Principle of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Instrumentation of Self-Sensing Permanent Magnet (PM) Motor . . . . . . . . . . . . . . . . . . . Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Smart Pantograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature Compensated Contact Force Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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T. Sun OBE () School of Mathematics, Computer Science and Engineering, City, University of London, London, UK e-mail: [email protected] M. Fabian · Y. Chen · M. Vidakovic · S. Javdani · K. T. V. Grattan · J. Carlton City, University of London, London, UK C. Gerada The University of Nottingham, Nottingham, UK L. Brun Faiveley Brecknell Willis, Somerset, UK © Springer Nature Singapore Pte Ltd. 2019 G.-D. Peng (ed.), Handbook of Optical Fibers, https://doi.org/10.1007/978-981-10-7087-7_19

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Abstract

Optical fibers have been explored widely for their sensing capability to meet increasing industrial needs, building on their success in telecommunications. This chapter provides a review of research activities at City, University of London in response to industrial challenges through the development of a range of optical fiber Bragg grating (FBG)-based sensors for transportation structural monitoring. It includes the instrumentation of marine propellers using arrays of FBGs mapped onto the surface of propeller blades to allow for capturing vibrational modes, with reference to simulation data. The research funded by the EU Cleansky programme enables the development of self-sensing electric motor drives to support ‘More Electric Aircraft’ concept. The partnership with Faiveley Brecknell Willis in the UK enables the integration of FBG sensors into the railway current-collecting pantographs for real-time condition monitoring when they are operating under 25 kV conditions. Keywords

Fiber Bragg Gratings · Marine propellers · Optical fiber sensors · Self-sensing electrical motors · Smart railway current-collecting pantographs · Structural condition monitoring

Fiber Bragg Grating (FBG)-Based Sensing Technology An intensive review of the use of fiber optics for structural condition monitoring has been undertaken (Grattan and Meggitt 1998) showing a number of techniques, amongst which the most widely used are Fiber Bragg grating (FBG)-based techniques (Kerrouche et al. 2009). FBGs produce wavelength encoded signals which are not susceptible to instrumental drift and environmental interference and have proven to be more robust and reliable, suitable for operation in harsh working conditions. Optical fiber Bragg gratings (FBGs) are used as a basis for simultaneous temperature and strain measurement. A FBG is a structure with the refractive index of the fiber core being periodically modulated and reflects the light at a wavelength termed the Bragg wavelength (B ) that satisfies the Bragg condition, given in Eq. 1 B D 2neff ƒ

(1)

where neff is the effective refractive index of the fiber core and ƒ is the grating period, where both are affected by strain and/or temperature variations, a feature that is reflected in the sensor design. The underpinning sensing mechanism of a FBG is that its Bragg wavelength (B ) shift is determined by the change in surrounding temperature and/or strain applied as described in Eq. 2 (Pal et al. 2005).

46 Optical Fiber Sensors for Remote Condition Monitoring of Industrial Structures

B D .1  P e/ " C Œ.1  P e/ ˛ C  T B

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(2)

where Pe is the photoelastic constant of the fiber, " is the strain induced on the fiber, ˛ is the fiber thermal expansion coefficient and  is the fiber thermal-optic coefficient. The first term of Eq. 2 represents the longitudinal strain effect on the FBG and the second term represents the thermal effect, which comprises a convolution of thermal expansion of the material and the thermal-optic effect. Equation 2 also indicates clearly the cross-sensitivity of a FBG to strain and to temperature, therefore when strain measurement is required using a FBG, its temperature effect is required to be compensated through optimizing the sensor design and sensor packaging. One of the key features that FBG-based sensors have demonstrated is their multiplexing capabilities. Fig. 1 shows a typical FBG-based sensor layout based on wavelength-division-multiplexing (WDM) (Othonos and Kalli 1999) with each grating (sensor point) being encoded with a specific wavelength. This characteristic is of particular importance for monitoring large-scale and/or critical structures that require densely distributed sensors, for simultaneous multi-point multi-parameter measurement yet with limited number of fibers (‘wires’). Compared to conventional strain gauge-based techniques, the FBG-based quasi-distributed sensing approach has shown significant advantages in terms of the ease of handling/installation and integration of a large number of sensing points, i.e., FBG strain/temperature sensors, coupled to a single source and interrogated by a single detector. In addition, there is no need to post-process the FBG raw data obtained due to their high signal-to-noise ratio compared to those from strain gauges. The WDM scheme, illustrated in Fig. 1, can be used very effectively to address a number of gratings, yielding not just the strain/vibration/force and temperature values of multiple sensors but also, through prior calibration, their physical locations on the target structures.

Fig. 1 Quasi-distributed FBG sensor system using wavelength-division-multiplexing technique

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FBG-Based Sensors for Monitoring Full-Scale Marine Propellers Background For marine propellers, their blades rotate slowly compared for example to blades in fans, jets and in turbines and hence, the centrifugal forces on the blades and therefore the stiffness increase that would result can be ignored, as it is minimal. In a series of experiments and subsequent calculations carried out by Conn (1939), it was concluded that it is mainly the flexural frequencies that are affected by the centrifugal forces. The flexural frequencies of a rotating blade can be calculated from the following relationship: f 2 D f02 C k2

(3)

where f is natural frequency of the rotating blade (Hz), f0 is the natural frequency of a non-rotating blade,  is the angular velocity (in revolutions per second (rps), this normally having a value of around 5–10 in the case of marine propellers) and k is a constant which has the values 0.35 and 1.35 for vibration parallel and perpendicular to the blade breadths. It can be seen from Eq. 3 that the effect of rotation on the frequencies of the blades is negligible. This was confirmed in work carried out by Castellini and Santolini (1998), where the natural frequencies of a smallscale, model propeller underwater were measured using a non-contact tracking laser vibrometer. They concluded that under rotating conditions, the bending modes were observed to be the most important vibration modes, since excitation due to hydrodynamic effects, gas bubbles or cavitation induces hardly any torsion effects on the blade structure. The principal effect of immersing a propeller in water is the reduction in the frequencies at the particular mode at which the vibration occurs. However, this reduction is not constant and appears to be greater for lower modes of vibration, when compared to higher modes. In order to investigate this effect, Carlton (2012) has defined the frequency reduction ratio as ƒD

frequency of mode in water frequency of mode in air

(4)

Considering a blade as a system with a single degree of freedom, the relationship between the motion of such a system under an undamped situation (while the stiffness remains unchanged) can be presented as a simple mass ratio equation, as shown in Eq. 5: Â ƒD

Mb Mb C Mw

Ã1=2 (5)

where Mb is the equivalent mass of the blade and Mw is the added mass of water.

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This research explores the potential of using a FBG-based sensor network for direct measurement of natural frequencies and mode shapes of vibrations of a full-scale marine propeller of modern design in water and in air. Compared to the conventional methods using accelerometers or strain gauges, this FBG-based approach shows significant advantages in terms of being minimally invasive on the blade itself and its performance, yet allowing for a very large number of sensing points to be investigated simultaneously. There is no electrical hazard with the use of these sensors in air, or more particularly in highly conductive sea water, unlike the case with electronic strain gauges. Compared to the point-to-point laser-based method, this approach is advantageous as it is insensitive to the refractive effect arising from directing a laser beam with high precision to a specific part of the surface of blade – indeed there is no need for the blade to be visible to the operator.

Experimental Setup The propeller selected for investigation in this experiment is a left-handed propeller designed for a twin-screw ship. The fixed pitch propeller blades had a diameter of 1900 mm with a variable pitch distribution of the blade. Table 1 presents the principal characteristics of propeller blade geometry and the material properties of the propeller. Figure 2a shows a photograph of the propeller immersed in a water tank of base size 4  4 meters (and wall height of 2 meters) and the 5 blades of the propeller are instrumented with a total of 335 sensing points with each blade being mapped with 67 sensors and their locations are pre-determined by simulation results as shown in Fig. 2b. All the FBGs used in this work are fabricated at City with an excimer laserbased fabrication system using phase mask technique. In each channel seven FBGs with different wavelengths, set to be between 1525 and 1565 nm, were designed and fabricated to ensure that there was no spectral overlap from one sensor to the next, even when each sensor responds over its maximum range of vibration-induced strain to avoid any ambiguity in the measurement. These FBG-based channels then formed the network of sensors on each blade as shown in Fig. 2b – a pattern that was repeated for each of the five blades of the propeller. The sensor location however, as illustrated in Fig. 2b, is designed to be as far as possible normal to the chordal lines, to enable the results obtained from any

Table 1 Characteristics of propeller under test

Diameter Mean pitch Expanded area ratio Modulus of elasticity, E Poisson ratio,  Density, ¡

1900 mm 1631 mm 0.765 121 GPa 0.33 7650 kgm3

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Fig. 2 (a) Left-handed propeller, instrumented with 335 sensors, in water tank; (b) an expanded view of a typical blade showing the sensor location points (numbered) used for the optical sensor vibrational analysis

individual sensor to be as closely comparable to those obtained from the other sensors. The modes of vibration of the blades, in air, were excited by striking the blades at various known locations on the propeller with a hammer with a relatively hard rubber tip, in order to excite the vibration of the blade across the full vibrational frequency range. Since the sensing interrogator unit was able to read data from 4 fibers simultaneously, first the optical fiber array next to the trailing edge of

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the blades was used as a reference for normalizing the captured amplitudes of the strain data. Therefore, when capturing data in this way, a series of three tests was necessary to obtain all the data for all the sensing locations for the blade shown in Fig. 2a. This was not difficult to do quickly and reproducibly. Then the amplitudes determined from the data from each set of tests were normalized to those obtained from the 1st sensor of the first fiber array, to allow the amplitude then to be comparable. Mode orders were tracked using a strain mode shape based (SMSB) method. Hence the strain mode shapes were generated using the normalized data obtained from each sensor location and the results were mapped onto an expanded surface of the blade and cross-compared with data obtained from across the different blades. In order to compare the experimental results with those from a simulation carried out, these results were then compared to those obtained from an extensive Finite Element (FE) analysis (Javdani et al. 2016), carried out using Abaqus software.

Experimental Results and Discussions Both the theoretical simulation and the experimental vibration tests in water were undertaken to determine the required water level to be used above the propeller blade, so that the natural frequencies are not affected by the depth of immersion. A deep tank was available and a series of preliminary tests was carried out using different water levels, namely 1350 , 1450 and 1550 mm, measured for convenience from the bottom of the water tank. The propeller in these tests was placed on a solid base of height 500 mm from the bottom of the tank, leaving the propeller blades to be immersed in 425 mm, 475 mm and 525 mm of water respectively from the free water surface. Further, tests were performed on the steady blades excited via the use of an impact hammer, in the conventional way. The results of these preliminary tests for a representative blade, Blade 3, are shown in Fig. 3. It can be seen from the results that increasing the water level from 1350 mm to 1550 mm does not affect the natural frequencies measured. Therefore it was considered unnecessary to increase the water level further and further tests in the tank were conducted using a 1350 mm water level. A direct comparison was made of the results of the simulation and the experimental measurements, for each blade and for air and water, looking at the natural frequencies of vibration of the blade. Figure 4 shows a direct comparison of one of these representative sets of experimental and simulation results – this being obtained for blade 3 and acting as an illustration. The results show an excellent agreement for the first 12 natural frequencies, illustrating fully the capability of the specially designed optical FBG-based sensor network in capturing the vibration behaviour of a complex structure, such as a marine propeller. This has been done over a wide frequency range, at multiple positions, and both in air and water. Such a great level of details cannot be easily achieved with conventional sensors and under water.

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600 Frequency (Hz)

500 400 300 200 100 0 1350 mm

1 52

2 3 4 5 6 7 8 9 10 137 151 250 286 406 419 426 495 568

1450 mm

52

137 149 251 286 408 419 425 494 567

1550 mm

51

137 150 251 287 408 420 426 495 568

Fig. 3 Natural frequencies measured at different water levels (shown below the x-axis). Natural frequencies (Hz) up to the 10th mode (the top numbers on the x-axis) are presented in the table, for three different water levels, 1350, 1450 and 1550 mm from the tank bottom

Fig. 4 Natural frequencies obtained from a series of tests, both experimental and simulation, both in air and in water, for blade 3

Conclusion As detailed above, a FBG-based sensor network provides a novel solution to acquire accurate information from arrays of sensing points for better analysis and thus better understanding of the displacement mode shapes of a full-sized actual marine propeller. This allows monitoring in real-time of the associated Bragg wavelength shifts from arrays (networks) of FBG sensors. The data obtained have a shown a very good agreement with those obtained from simulation using finite element techniques, within experimental error. It was also confirmed that the location of the excitation on the blades, as expected, directly affected the amplitudes of the various frequencies that were detected in the experimental work.

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Self-Sensing Electric Motor Introduction The ‘more electric’ concept in many areas of engineering has increased significantly the demand on reliability, power density and manufacturing efficiency of rotating electrical machines. To address this ever growing demand for new and reliable designs, electrical machines are increasingly required to be monitored in real-time with the data obtained being used for both model validation and prototype diagnostics. The latter helps to identify potential modes of failure and thus ensure the drive’s reliability, as requested by machine owners or end users. If a conventional approach were to be adopted to achieve such multipoint, multi-parameter measurements, it would involve a drastic increase of component count thus reducing the overall reliability of the system in question. Further to this, due to the relatively large size of insulated conventional sensors, the resulting system will potentially occupy a spatial envelope larger than the drive itself. This work thus aims to address the above challenges, by replacing such conventional sensors with an integrated optical fiber-based, quasi-distributed, sensing system in electrical machines for real-time monitoring. Such a novel approach takes full advantage of the fiber sensors’ reduced spatial envelope and immunity to electromagnetic interference. One of the first efforts made in the direction of introducing an optical fiber sensor for motor and drive applications was to exploit Rayleigh backscattering in conjunction with a fiber having its outer cladding modified at intervals for a quasi-distributed temperature measurement system using an optical time domain reflectometry (OTDR) (Boiarski and Kurmer 1997). Since then different optical sensing techniques were applied to monitor end-winding vibratory behaviour (Kung et al. 2011), stator housing vibration (Corres et al. 2006), thermal effects (De Morais Sousa et al. 2012) and torque (Swart et al. 2006), for instance. In a previous report, the authors introduced a stator wave and rotor speed tracking system based on fiber Bragg grating (FBG) sensors (Fabian et al. 2015). This research aims to explore the ‘all-in-one’ sensing concept underpinned by the FBG technique for simultaneous measurement of all the key parameters required, and operated by a single sensing interrogation unit thus eliminating the need for individual sensor systems for each parameter and therefore significantly reducing the complexity of electrical machine condition monitoring.

Principle of Operation In order to evaluate the all-in-one sensing concept a PMAC machine was instrumented with a total of 48 FBGs at specific locations within the motor. The principle exploited to measure vibrations, the rotor speed and its position, the stator wave frequency and the spinning direction is based on the spatial modulation of the air-gap flux in the stator core of an induction machine. The resulting stator teeth displacement can be measured in the form of strain using FBGs as previously

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reported by the authors (Fabian et al. 2015). The method employed to measure torque is based on a differential wavelength approach where two FBGs are attached to the rotor shaft at an angle of ˙45ı with respect to the spinning axis (Swart et al. 2006). In this configuration, the difference between the two FBG reflection peak wavelengths is a measure for torque whereas their mid-point is an indicator for the temperature at that location. The dynamic Bragg wavelength shifts of all 48 FBGs were captured simultaneously using a Micron Optics SM130 sensing interrogator unit, at a sampling rate of 2000 Hz. The DC components of the transient signals were used for thermal analysis (and torque) whereas the AC components were used to determine stator vibrations and phase shifts necessary extract the dynamic parameters.

Instrumentation of Self-Sensing Permanent Magnet (PM) Motor A self-sensing PMAC electric motor, instrumented with 48 FBG sensing points with 36 installed in the stator and 12 on the rotor, is shown in Fig. 5. The connection between the sensing fiber integrated into the rotor and the FBG interrogator is achieved using a fiber-optic rotary joint via a shaft adapter created using a 3-D printer. The instrumentation of the stator is as follows. Two fibers of 12 FBGs each were routed along the stator windings for thermal profiling, two FBGs in each stator slot as shown in Fig. 6a, b. The fibers were looped around several times at either end of the stator core. A third fiber of 12 FBGs was circumferentially mounted on the stator core with each FBG placed in between adjacent stator teeth (Fig. 6c) to measure vibrations, the rotor speed and its position, the stator wave frequency and the spinning direction.

Fig. 5 PM machine instrumented with 48 FBGs on 4 fibers measuring rotor speed, torque, vibration, the rotor magnet temperatures and stator end-winding temperatures

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Fig. 6 Schematic (a) and photograph (b) of the distribution of 24 FBGs along the stator windings for their thermal profiling. (c) Schematic of the 12 circumferentially mounted FBGs used to measure stator vibrations that result in a number of parameters possible to be extracted

The rotor instrumentation is as follows. The remaining 12 FBGs were distributed across the rotor surface, one on each of the ten magnets, again for thermal profiling, and the other two on the rotor shaft for simultaneous torque and temperature monitoring. This is illustrated in Fig. 7. The rotor fiber was interrogated by means of a fiber-optic rotary joint which allows for the continuous monitoring of the rotor condition while spinning. After the completion of the instrumentation and PMAC motor assembly discussed above, the FBGs used for temperature monitoring were calibrated by putting the motor in a climate chamber, as shown in Fig. 8, and running a pre-programmed temperature cycle (20–70 ı C in steps of 10 ı C). The collected data from each FBG at each temperature step were averaged and then individually fitted using a linear least squares algorithm as shown in Fig. 9. Figure 9a shows the temperature-dependent Bragg wavelength shifts of the 34 FBGs (24 in the stator and 10 the rotor), used for rotor and stator temperature profiling, at the temperature intervals mentioned above. It can be observed from Fig. 9a that the FBGs attached to the stator end-windings exhibit approximately

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Fig. 7 Schematic and photograph of the instrumented PMAC rotor. One fiber of 12 FBGs is attached to the rotor, one to each of the 10 rotor magnets and 2 FBGs on the rotor shaft for torque measurement

Fig. 8 Temperature calibration of the integrated FBG sensors in a climate chamber

three times the wavelength shift as the ones attached to the rotor magnets. This is due to the different thermal expansion coefficients of the end-winding material and the magnet. Figure 9b shows the fitted curves of two exemplary FBGs, with one installed on the end-winding of the stator and other on the rotor, to highlight their different temperature sensitivities (30 pm/ı C for the stator FBG vs. 10 pm/ı C for the rotor FBG). It is also clear from the fitted curves that the temperature induced Bragg wavelength shifts are highly linear. With this information known, the absolute temperature T can be derived from the monitored Bragg wavelength B of an FBG using Eq. 6 which is modified from Eq. 2, where cT is the temperature coefficient or sensitivity of the FBG in nm/ı C, obtained through calibration, and T0 is the˙45 T0 ı C. T D

1 .B  T 0 / cT

(6)

The test bed was set up at the Institute for Aerospace Technology on the University of Nottingham’s Jubilee Campus. The load motor was purchased from

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Fig. 9 Temperature calibration curves. (a) Bragg wavelength shift of all 34 FBGs used for temperature profiling; (b) Linear fitting of the temperature-dependent Bragg wavelength shifts of two exemplary FBGs

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Fig. 10 Photograph of the assembled test bench and corresponding instrumentation hardware at the Institute for Aerospace Technology, Nottingham University

City Rewinds and Drives, and its corresponding 55 kW drive cubicle from Emerson Industrial Automation which also includes the drive for the PM machine. The torque transducer and couplings were sourced from Magtrol and smaller bits and pieces (encoder, cabling, bolts, etc.) from standard suppliers in the field. The test bench, as shown in Fig. 10, was thus assembled and commissioned by the technical team at the University of Nottingham who also made the shaft guard. As shown in the figure, the test bench includes the PMAC motor drive under test and load drive, integrated with both optical and conventional sensors.

Results and Discussions Figure 11a shows the frequency response of one of the circumferentially mounted FBGs with the machine spinning at 16.7 Hz excitation. The first spectral feature at 16.7 Hz represents the rotor speed and the second (main) feature at 167 Hz corresponds to the stator wave frequency. Either of the two can be used to extract the rotor speed and convert it to rotations per minute (rpm). Other vibrational information, as evident from Fig. 11a, gives machine developers and engineers an important insight into the vibratory behaviour of a machine’s design. Since vibrations are also an early indicator for impending machine failure, the constant monitoring of vibrations is of high importance in increased reliability environments. Figure 11b shows the rotor speed obtained from the FBG data against a reference sensor with the rotor speed being varied between 1000 and 1600 rpm. It is clear from

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Fig. 11 (a) Frequency response of one of the circumferentially mounted FBGs in the case of a spinning rotor under no load. (b) Rotor speed versus time obtained from the FBG sensor and from a reference sensor

Fig. 10b that the FBG approach very closely matched the reference sensor signal at a much improved signal-to-noise ratio. Figure 12a shows the dynamic responses of four of the circumferentially mounted FBGs highlighting the phase shift between them. This phase shift can be used to track the rotor position with regard to a reference point, i.e., acting like a conventional encoder. The phase shift between any two of those FBGs can also be used to determine the spinning direction of rotor, a positive phase shift indicating rotation in one direction and a negative phase shift rotation in the other direction. Figure 12b shows the differential mode wavelength shift (the distance between the two FBG reflection peaks) of the torque sensor layout at varied levels of torque up to 2 Nm. Again, a very close correlation between the FBG approach and the reference sensor has been achieved with a linear torque – wavelength shift relationship (21.5 pm/Nm). In practise it is challenging to realise an angle of

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Fig. 12 (a) Exemplary phase shifts of the four of the circumferentially mounted FBGs. (b) Differential mode wavelength shift of the FBG torque sensor compared with the data of a reference torque transducer

exactly 90ı between the two FBGs meaning that the differential mode wavelength will experience some sort of temperature dependence. However, this is easily compensated for using the torque-independent mid-point wavelength as an indicator for the temperature. Figure 13 shows a typical screenshot of the motor monitoring application GUI at runtime. The top left graph shows the stator end-winding temperatures versus time and the rotor temperatures are shown in the graph below it. The temperature data are also visualised on the right in the form of colour coded 3-D models of both the stator and the rotor. The two graphs in the centre show the torque data and rotor speed. The latter is extracted from the spectral response of one of the circumferentially mapped FBGs. The vibration signature of the machine is shown in the bottom graph. In the shown case, the first spectral feature corresponding to the mechanical rotor frequency (rotor speed) and the second feature to the stator wave frequency. The FFT window length can be changed with the resulting frequency resolution being shown above the graph. The buttons in the top right to pause the data acquisition, to log the data and to terminate the application and are self-explanatory. In addition

Fig. 13 Screenshot of the motor monitoring application at runtime showing key motor parameters

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Fig. 14 Sensor data mapping in software through 3D models of rotor and stator. (a, b) Localised heating of fibers of the same sensor layout as the actual ones attached to the rotor and stator end-windings to evaluate the mapping algorithm. (c) Sensor mapping configuration window in LabVIEW™

to the data logging function, the contents of each graph can be exported to Excel at runtime with only two mouse clicks. This GUI design has been made flexible and can easily be edited by the end user to accommodate possible changes in the testing environment required by industry. Figure 14 shows temperature data mapped in a 3-D format for both the rotor and stator as included in Fig. 13. In the mapping configuration window of Fig. 14, the sensor locations were marked on the model surface and a 1-D array containing the temperature data is wired to the VI. The algorithm developed has been validated by using fibers of identical FBG layout as the ones used in the motor that were locally heated. Figure 14a, b show the resulting screenshot of the 3-D real-time visualisation. The data obtained are also managed and prepared for the next-stage integration into the motor control software.

Conclusion It was shown that when placing a network of FBGs at certain locations within an electrical machine, comprehensive condition monitoring can be performed at

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a high level of accuracy. Multiple parameters can be extracted from the FBG data by using appropriate data processing and compensation algorithms. The proposed all-in-one sensor system has the potential to replace conventional systems that require a separate sensor/system for each parameter to be monitored. It reduces the component count and spatial envelope of a test environment as all sensing elements are confined within the machine with minimum external wiring/coupling as opposed to conventional sensors. Future work will focus on the implementation of an active feedback control system using the sensor data to control a machine’s speed under different load conditions, for instance.

Smart Pantograph Introduction The pantograph is a critical, roof-mounted part of a modern electric train, tram or electric bus to collect power through an overhead catenary wire and successful current collection requires a reliable pantograph-catenary contact with a steady force under all conditions, as the train travels along the line. The pantograph operates in a particularly harsh environment, being exposed to all weathers as its carbon strip rubs along the OLE at speeds up to 125 mph and at 25,000 volts conditions: monitoring its condition in real-time has posed a real technical challenge to the rail industry and optical fiber sensing provides an effective solution. This research exploits fully the key advantages of FBGs and their suitability for pantograph condition monitoring, in terms of their immunity to electromagnetic interference, ease of multiplexing, small size and lightweight. The major drawback, however, of using a FBG-based technique is its cross-sensitivity to strain and to temperature, therefore a significant amount of effort has been made to compensate the temperature effect when a FBG is used for strain measurement. Camolli et al. (2008) reported the use of two single FBG sensors on separate fibers, where one of the FBGs is used for temperature compensation. This approach is based on the assumption that the temperature distribution is uniform along the pantograph, however this is not necessarily the case in a real time situation. The other FBGbased sensor system (Wagner et al. 2014) deploys the use of aluminum boxes confining 3 FBG sensors within a small footprint, with one strain-free FBG for temperature compensation. Each pan-head is instrumented with two boxes which increases the mass and consequently affects aerodynamic force when the train moves at high speeds. Embedding the FBG sensors between carbon and aluminum has been reported by Schroder et al. (2013). All the reported FBG techniques require either an additional fiber or an additional FBG for temperature compensation. Considering the high temperature sensitivity, which is one order of magnitude higher than that of its strain sensitivity, it is challenging to remove the temperature effect in a satisfactory way and this forms the core of this research.

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Fig. 15 Smart pantograph integrated with an FBG array

Temperature Compensated Contact Force Measurement Figure 15 shows a pantograph integrated with arrays of FBGs, which have been designed and configured to allow both real-time measurement of the contact force and contact location and facilitate a closed-loop control to avoid unexpected failure of pantograph during operation. This is a joint development between City University of London and Faiveley Brecknell Willis in the UK. This research exploits a novel temperature compensation method by using a package-based sensor design where three FBG-based sensor packages are integrated into three different locations of a pantograph, both for real-time measurement of the contact force and contact location and for temperature compensation, as illustrated in Fig. 15. Given the small footprint of each package, it is observed that the FBGs in the same package experience the same scale of temperature variations. This effect has been exploited in this research for effective temperature compensation. To verify the above sensor design idea, Fig. 16 shows an experimental setup created for the evaluation of the developed temperature compensation method for the contact force measurement under high current conditions. As shown in the figure, a current supply was connected to both sides of the pantograph, allowing for a step change of current from 0 to 1500 A and then from 1500 A to 0 A to be applied. The higher current applied induces temperature change in the range from 25 ı C to 55 ı C over a period of 9 m, as recorded by three thermocouples, which were co-located with 3 FBG packages as shown in Fig. 16. To speed up the cooling process, a fan is used. During the whole measurement process, there is no contact force applied to the pantograph.

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Fig. 16 Pantograph driven by a DC current, changing from 0 to 1500A and then from 1500 to 0A, when the contact force is zero

Fig. 17 Temperature change recorded by the central thermocouple and current value applied to the pantograph during the experimental tests

Experimental Results and Discussions Figure 17 shows the current applied and the temperature change recorded by the thermocouple which is co-located with the central FBG package. The current supply is switched off when 55 ı C is reached and a fan is used to accelerate the cooling process. Figure 18a shows the wavelength shifts of three FBGs, i.e., FBG4 , FBG5 and FBG6 , confined in the central package and located at the central area of the pantograph when a step change of current is applied to the pantograph from 0 to 1500 A and then from 1500 A to 0 A. The wavelength shift of each FBG within the same package experiences the change in applied strain (from the contact force)

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Fig. 18 (a) Wavelength shifts of FBGs 4, 5, 6 in the Central package recorded during the experimental tests; (b) The contact force measured by FBG packages before and after temperature compensation

and in temperature. The latter however can be subtracted based on the FBGs within the same package under the temperature variation and this underpins the algorithm developed for temperature compensation at City University of London. The black curve in Fig. 18b shows the contact force calculated from the data recorded by the central FBG package without considering temperature compensation. The red curve in Fig. 18b, however, shows clearly the contact force to be zero after the implementation of the temperature compensation algorithm developed and this agrees well with the test condition as the pantograph was electrified but without being in contact with the overhead line equipment (OLE). It is noticeable that temperature compensated FBG-sensor system removes the effect from the temperature changes and provides the information of the force being unaffected by the applied current.

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Further to the above temperature-compensated contact force measurement using each FBG package, the contact location of the OLE against the pantograph can thus be obtained by calculating the ratio of the contact forces measured simultaneously by the three FBG packages integrated into the pantograph at three different, yet known locations.

Conclusion This research explores a novel sensor design through integration of FBG sensor packages into a railway current-collecting pantograph for its remote condition monitoring. This is designed to remove temperature effect in the strain/force measurement and under the circumstances that the temperature effect is more dominant. The positive outcomes obtained from the field tests, by driving the pantograph using high currents, have further confirmed the effectiveness of this method used for temperature compensation. The research is still on-going and more vehicle tests will be undertaken in the near future to evaluate extensively the smart pantograph developed in industrial settings.

Summary A range of FBG-based optical fiber sensor systems have been developed and evaluated, showing promise for wide industrial applications. These sensors have been designed to provide real-time measurement of a suite of key parameters that would help engineers to diagnose structural conditions thus to improve structural integrity and reliability through improved maintenance.

References A.A. Boiarski, J.P. Kurmer, in Electric Power Research Institute, TR-101950-V2, 2487-02, Final Report (1997) J.S. Carlton, Chapter 21, in Propeller Blade Vibration in Marine Propeller and Propulsion, 3rd edn., (Butterworth-Heinemann, Oxford, 2012), pp. 421–429 P. Castellini, C. Santolini, Measurement 24(1), 43–54 (1998) L. Comolli, G. Bucca, M. Bocciolone, A. Collina, in Proceeding of SPIE, vol. 7726 (2008) J.F.C. Conn, in Proceedings of the Trans. IESS, 225–255 (1939) J. Corres, J. Bravo, F.J. Arregui, I.R. Matias, IEEE Sensors J. 6(3), 605–612 (2006) K. De Morais Sousa, A.A. Hafner, H.J. Kalinowski, J.C.C. Da Silva, IEEE Sensors J. 12(10), 3054–3061 (2012) M. Fabian, J. Borg Bartolo, M. Ams, C. Gerada, T. Sun, K.T.V. Grattan, in Proceedings of the SPIE 9634 (2015), 963417 4 pp S. Javdani, M. Fabian, J.S. Carlton, T. Sun, K.T.V. Grattan, IEEE Sensors J. 16(4), 946–953 (2016) A. Kerrouche, W.J.O. Boyle, T. Sun, K.T.V. Grattan, Sensors Actuators A Phys. 151(2), 107–112 (2009) P. Kung, L.Wang, M.I. Comanici, in Proceedings of the IEEE Electric Insulation Conference (2011), pp. 10–14

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K.T.V. Grattan, B.T. Meggitt (ed.), Optical Fiber Sensor Technology, vol. 3 (Kluwer, London, 1998). ISBN:978-1-4419-4736-9 A. Othonos, K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing (Artech House, Boston, 1999) S. Pal, Y. Shen, J. Mandal, T. Sun, K.T.V. Grattan, IEEE Sensors J. 5(6), 1462–1468 (2005) K. Schroder, W. Ecke, M. Kautz, S. Willett, M. Jenzer, T. Bosselmann, Opt. Lasers Eng. 51, 172–179 (2013) P.L. Swart, A.A. Chtcherbakov, A.J. Van Wyk, Meas. Sci. Technol. 17(5), 1057–1064 (2006) R. Wagner, D. Maicz, W. Viel, F. Saliger, C. Saliger, R. Horak, T. Noack, in 7th European Workshop on Structural Health Monitoring (2014)

47

Optical Fiber Sensor Network and Industrial Applications Qizhen Sun, Zhijun Yan, Deming Liu, and Lin Zhang

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ultra-Weak Fiber Bragg Grating (UWFBG) Sensor Network and Applications . . . . . . . . . . TDM-Based Quasi-Distributed Sensor Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TDM-Based Continuous-Distributed Sensor Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . WDM/FDM-Based Quasi-Distributed Sensor Network with High Spatial Resolution. . . Quasi-Distributed Sensor Network Based on 3D Encoded Microstructures . . . . . . . . . . . Special Fiber Grating Sensor Network and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tilted Fiber Grating (TFG) Sensor Network and Applications for near Infrared Detection (NID) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Er3C Doped Fiber Grating Sensor Network and Applications . . . . . . . . . . . . . . . . . . . . . . Fiber Optic Sensors Passive Optical Network (SPON) and Applications . . . . . . . . . . . . . . . TDM-Based Fiber Optic Acoustic SPON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . WDM/TDM-Based Fiber Optic SPON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

For many of sensing applications, multiplexed sensor networks which can map the sensing signal of a large structure or surveying at complex conditions are required, greatly promoting the development of the fiber optic sensor network with large capacity. In this chapter, three typical fiber optic sensor networks

Q. Sun () · Z. Yan · D. Liu School of Optical and Electronic Information, Next Generation Internet Access National Engineering Laboratory (NGIAS), Huazhong University of Science and Technology, Wuhan, Hubei, P. R. China e-mail: [email protected] L. Zhang Aston Institute of Photonic Technologies, Aston University, Birmingham, UK © Springer Nature Singapore Pte Ltd. 2019 G.-D. Peng (ed.), Handbook of Optical Fibers, https://doi.org/10.1007/978-981-10-7087-7_20

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and their applications will be introduced. Firstly, the ultra-weak fiber Bragg grating (UWFBG) sensor networks with ultra-large capacity for quasi-distributed and continuous distributed sensing in a single fiber link are investigated, which is realized by the multiplexing of UWFBGs or UWFBG based Fabry-Parot interferometers (FPI). Secondly, special fiber grating sensor networks with advanced functions and competitive performances, including the tilted fiber grating (TFG) sensors or distributed Bragg grating fiber laser (DBRFL) sensors multiplexed in a single fiber, are investigated. Thirdly, fiber optic sensors passive optical networks (SPON) with good adaptability, high extendibility, and great flexibility are comprehensively studied, which includes the star topology SPON and the tree topology SPON for colored sensors and colorless sensors accessing. For each type of sensor network, the sensor structures, networking mechanisms, system architectures, demodulation methods, and typical sensing performances are systematically discussed. Moreover, the developed systems or equipment and field tests for a wide range of commercial and industrial applications, especially for resource exploration, geophysics, infrastructure, medical diagnosis, food quality, and security control, are presented. Keywords

Fiber optic sensor network · Fiber grating sensor network · Fiber optic distributed sensing · Senor passive optical network

Introduction Due to the distinct advantages of light weight, small size, high sensitivity, immunity to electromagnetic interference, and ease to network, there is a high demand for smart optical fiber sensor technologies due to increasingly application needs in a wide range of sectors, including civil engineering, aerospace, maritime, energy, and defense industries, as well as in medical, environmental, and food sectors. Recent market analysis by ElectroniCast has reported that the global market value for fiber optic sensors was projected to $3.38 billion in 2016 and will increase to more than $5.98 billion in 2026 (ElectroniCast consultants 2017). For many of sensing applications, multiplexed sensor networks which can map the sensing signal of a large structure (e.g., oil-gas well, pipeline, bridge, border, aircraft wing, etc.) or geophysical surveying at complex conditions are required, for which single or pairs of sensors are not sufficient. Therefore, the fiber optic sensor network with large capacity is becoming an inevitable tendency for the sensing industry. The fiber optic sensor network mainly includes point fiber sensor array and distributed fiber sensor system, of which the sensor units can be multiplexed by specific schemes, including time division multiplexing (TDM), wavelength division multiplexing (WDM), frequency division multiplexing (FDM), space division multiplexing (SDM), or their combinations. Apart from seeing many successful commercial deployments of fiber sensors, novel and function-enhanced fiber sensors have been developed by utilizing specially modified structures and speciality fibers.

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In this chapter, three typical fiber optic sensor networks will be introduced, including the ultra-weak fiber Bragg grating (UWFBG) sensor network (section “Ultra-Weak Fiber Bragg Grating (UWFBG) Sensor Network and Applications”), special fiber grating sensor network (section “Special Fiber Grating Sensor Network and Applications”), and fiber optic sensors passive optical network (SPON) (section “Fiber Optic Sensors Passive Optical Network (SPON) and Applications”). The sensor structures, networking mechanisms, system architectures, demodulation methods, and the sensing performances will be discussed. Moreover, a wide range of industrial applications, especially for resource exploration, geophysics, infrastructure, medical diagnose, food quality, and security control will be presented.

Ultra-Weak Fiber Bragg Grating (UWFBG) Sensor Network and Applications Over the last decade, one of the most versatile and broadly researched and developed optical fiber sensor platforms is the in-fiber gratings, owing that the modulation pattern of the refractive index (RI) in fiber grating is sensitive to external parameters such as temperature, strain, and surrounding RI, resulting in the nominal wavelength shift. Until now, fiber Bragg gratings written on standard fiber have been widely used for measuring temperature, strain and force, pressure, vibration, liquid level, displacement, twist and torsion, bending and loading, current and magnetic field, chemicals and biochemical, etc., which are showing great potential and broad market prospects in industrial fields. The wavelength encoded nature of the information facilitates WDM for sensor networking, achieved by assigning individual sensors to a different slice of the available source spectrum (Fallon 2000), as illustrated in Fig. 1. This outstanding advantage makes fiber gratings become ideal candidates for many applications. Except for WDM, FBG can also be multiplexed by TDM or the combination of them to build a sensor network along one fiber for large area measurement.

Fig. 1 Schematic of the fiber grating sensor network

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However, the insertion loss induced by number of the FBGs is high, and therefore, the multiplexing capacity is limited to be about several hundreds, typically less than 100. Recently, UWFBG -based sensor network was proposed and investigated. The reflectivity of FBG is smaller than 1%, even decreased to about 105 , so the insertion loss into the fiber can be ignored. Consequently, there is enormous potential in the multiplexing capacity along single fiber. In this section, four types of UWFBG-based sensor network and the typical applications will be discussed.

TDM-Based Quasi-Distributed Sensor Network Based on the narrow bandwidth and weak reflectivity of UWFBG, identical gratings can be multiplexed in time domain to improve the multiplexing capacity and sensing distance greatly. Figure 2a presents the working principle of the identical UWFBG multiplexed sensor network for quasi-distributed sensing. When pulse signal incidents into the fiber, a pulse sequence will be reflected and different pulse corresponds to different UWFBGs. Meanwhile, the central wavelength of the reflected pulse presents the value of the sensing parameter of it. It means that the UWFBG sensors can be interrogated in both time domain and wavelength domain, achieving multi-point synchronous precision measurement and positioning. The propagation of the optical wave in the fiber is similar with the backscattering of fiber, such as Rayleigh scattering, Brillion scattering and Raman scattering which can implement long distance and distributed sensing. Hence the sensing system is named as the microstructure-OTDR, i.e., M-OTDR, because the UWFBGs can be considered as longitudinal distributed microstructures. While the reflectivity of the UWFBG is several orders of magnitude higher than backscattering light as shown in Fig. 2b, it is a perfect candidate for improving the signal to noise ratio (SNR) of the backscattering light in fiber as the sensing point, resulting in the higher measurement precision and greatly shorter response time. Meanwhile, by networking the identical UWFBGs through TDM, the multiplexing capacity in a single fiber can be greatly improved to 1000 due to the relatively lower insertion loss (Zhang et al. 2012a). Along with the decrease of the UWFBG reflectivity, the effect of the cross-talk induced by multi-reflection between the gratings will be weakened gradually (Hu et al. 2014). When the UWFBG reflectivity is about 40 dB with the central wavelength of 1550.9 nm, the multiplexed number of gratings could reach up to1642, showing a low transmission loss (Wang et al. 2016). Owing to the advantages of large multiplexing capacity, high measurement accuracy, and long-sensing distance, this sensing network has a great potential for health monitoring of bridges, dams, tunnels, and other distributed sensing applications. For example, the network made up of 6108 UWFBGs with two wavelength bands in a 10 km fiber was developed and the distributed temperature measurement was conducted by using a temperature test chamber. The experimental results were shown in Fig. 3, exhibiting the red shift of peak wavelength with the increase of temperature, measurement accuracy of 0.5 ı C, and good linear response with the coefficient around 10.68 pm/ı C at any gratings (Yang et al. 2016).

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Fig. 2 Schematic of the identical UWFBG multiplexed sensor network: (a) working principle; (b) OTDR trace of the UWFBG array. (Copyright 2019 Springer)

TDM-Based Continuous-Distributed Sensor Network Although high SNR can be realized with UWFBG, fiber between two neighboring UWFBGs becomes dead zone of the sensing system. To detect the event occurred in this section, the UWFBGs based fiber as the sensing link and the coherent OTDR as the demodulation scheme were combined to realize wideband

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and high sensitive fully-distributed sensing. As illustrated in Fig. 4, light from the laser source is modulated into pulses and frequency shifted by f after passing through the acoustical optical modulator (AOM). After amplified by EDFA and filtering process, reflected pulses from the UWFBG are combined with the local oscillator and received by the balanced photo detector (BPD). The reflected signal contains the intensity and the phase information of every UWFBG. Phase change on the sensing fiber induced by external parameters is obtained by calculating the phase difference between every two adjacent UWFBGs through differential cross-multiplying (DCM) algorithm. And the event location is identified by searching the peaks of the backscattered light; therefore, the spatial resolution is determined by spatial interval of UWFBGs (Ai et al. 2017). Based on the above setup, distributed vibration, acoustic wave, strain, and temperature, detection were explored. Specifically, as the phase change induced by the temperature and the vibration event would occupy different frequency band, after the low pass filter (LPF) and the high pass filter (HPF), the vibration and temperature

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Fig. 4 Configuration of the distributed sensing system based on UWFBGs sensing link. (Copyright 2019 OSA)

change can be measured simultaneously (Ai et al. 2018). The experimental results are displayed in Fig. 5, which demonstrate that the sensor system can accurately map the temperature distribution and trace the multiple vibrations along the sensing fiber, as well as simultaneously detect the temperature change and vibration signal occurred at the same position of 752 m. In addition, the sensor network provides excellent sensing performance for distributed acoustic sensing (DAS) with wideband covering from static to ultrasonic range. When no strain was applied on the sensing fiber, phase noise spectral densities at low frequency and high frequency were recorded and presented in Fig. 6a, b. It is clear that the phase noise above 1 Hz is as low as 7  104 rad, which means that the system can respond to the acoustical signal at ultra-low frequency region. To evaluate the sensitivity to acoustical waves, static strain test was conducted. As shown in Fig. 6c, sensitivity of 4.393 rad/" as well as good linear relationship with R2 > 0.9998 were achieved. From the phase noise spectral density above, the strain resolution over 1 Hz can be deduced to be lower than 0.16 n". Meanwhile, the demodulated acoustic distribution along the sensing fiber was investigated. The experimental results in Fig. 7a, b prove that the DAS system owned wide response band from 0.1 Hz to 45 kHz, with successful recovery of the acoustic wave. It should be emphasized that not only the UWFBG with certain wavelength selection but also the ultra-weak chirped FBG(UWCFBG) with wideband selection or the local abrupt change point of RI without wavelength selection can be served as the backscattering enhanced microstructure, which are inscribed in the fiber through UV or femtosecond lasers exposure. The M-OTDR based DAS system have been widely applied in industrial fields such as borehole survey (Yamate et al. 2017; Mateeva et al. 2014), seismic recording (Ni et al. 2005; Jousset et al. 2018), and rail crack detection (Fan et al. 2019). Figure 8 presents our field test conducted in an oilfield (Fig. 8a). A 1 km long UWFBG array fiber cable was deployed into a cased borehole with a weight bar to pull the fiber cable down to the borehole (Fig. 8b). An explosive source was used to generate seismic energy on the surface with different

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offset distance to the wellhead. From Fig. 8c, it can be seen that the fiber DAS system based on UWFBG sensor network acquired borehole seismic data with good quality, where both of the upgoing wave and downgoing wave can be clearly detected. Figure 9 illustrates the field test of fiber DAS system to record seismic data. From Fig. 9a–c, it can be seen that the distance between the seismic signal and the seismometer or DAS system is about 8.7 km, and 500 m SNR enhanced sensing fiber cable was buried underground with the depth of about 20 cm to record the seismic wave transmission. Figure 9d clearly shows the excited wave by heavy hammer near the fiber, and the comparison of the recorded seismic data

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Fig. 6 Noise floor and sensitivity of UWFBG array fiber based DAS: (a) phase noise spectral density of low frequency less than 10 Hz; (b) phase noise spectral density from 0.1 to 80 kHz; (c) relationship between phase change and applied strain

from electrical seismometer and fiber DAS in Fig. 9f demonstrates that the fiber DAS can detect the seismic data with high sensitivity, high accuracy as well as wider response frequency band. In addition, as presented in Fig. 9e, the fiber DAS system successfully recorded the vehicle movements in this area all the time, which can used for analyzing the traffic condition and the noise pollution in the city. Figure 10 shows our application test for rail crack detection, where the UWFBG array fiber cable was laid on rail waist (i.e., in the middle of the rail track, which is also called rail web) (Fig. 10a). When the train went through a crack on the rail, a strong acoustic source was excited at this position owing to the interaction between the wheels and uneven rail, and then propagated both in forward and backward directions. As depicted in Fig. 10b, c, by analyzing the temporal and spatial distribution of sound waves recorded by the DAS system, the intersection point of the forward and backward wave propagation traces can be found, which corresponds to the accurate location of the crack. The method has great prospect in enhancing railway safety.

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Fig. 8 Field test of borehole survey: (a) photograph of the well site; (b) photograph of the borehole and fiber cable; (c) recorded borehole seismic data excited by explosive source

WDM/FDM-Based Quasi-Distributed Sensor Network with High Spatial Resolution The locating principle in TDM scheme described above is still based on the time delay tracking, so the spatial resolution is relatively larger than 1 m, which is not enough for special applications. In order to further increase the multiplexing capacity and improve the spatial resolution of the UWFBG sensor network, a fiber microstructure as shown in Fig. 11a was proposed and designed, which can be considered as Fabry-Pérot interferometer (FPI) composed of two closely spaced UWFBGs. Owing to the weak reflectivity of the gratings, one microstructure can be considered as a low-finesse FPI, of which the reflectivity RS can be simplified as a

Fig. 9 Field test of seismic recording: (a) google map of the test site, where the distance between the seismic signal transmitter and the fiber DAS system is about 8.7 km; (b) photograph of the seismic signal transmitter; (c) photograph of the shallow buried sensing fiber cable; (d) recorded seismic data excited by heavy hammer signal near the fiber cable; (e) recorded seismic data excited by vehicle movements; (f) comparison of the recorded seismic data from electrical seismometer and fiber DAS, which was excited by the distant seismic signal transmitter

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Fig. 10 Field test of railway monitoring: (a) photographs of the railway attached with fiber cable and crack on the track; (b) acoustic wave distribution along the railway measured by fiber DAS system; (c) Zoom of certain section in (b) after filtering. (Copyright 2019 OSA)

Fig. 11 (a) The configuration of the microstructure; (b) the reflection spectrum. (Copyright 2019 OSA)

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two-beam interferometer (Miridonov et al. 1998): RS D 2RG Œ1 C cos .4neff LC =/

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The backward scattering spectrum RS of the microstructure is an UWFBG envelop modulated by F-P comb filtering as depicted in Fig. 11b, which is related to the Bragg wavelength B and the frequency vc . Therefore, the microstructures can achieve wavelength and frequency encoding simultaneously (Zhang et al. 2019). The multiplexing capacity of the microstructures is analyzed for high spatial resolution distributed sensing, which can be calculated through the number of WDM channels and FDM channels as follows: N D NWDM  NFDM D

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Where ıB is the operating bandwidth of the microstructures, Lm and LC are respectively the maximum cavity length of the microstructure and the cavity length difference between the microstructures in adjacent FDM channels, and ıs is the bandwidth of the interrogation system. Theoretically, over 1000 microstructures with spatial interval less than 1 cm can be high-densely multiplexed along the single fiber (Li et al. 2012a). In order to achieve high resolution and fast response, a demodulation system is developed by combining the fiber Fabry-Perot tunable filter (FFP-TF) tuning with a parallel signal processing algorithm. As shown in Fig. 12, the parallel processing flow of demodulation scheme contains four parts. Firstly, the modulated optical signal from the sensor array is scanned using a FFP-TF controlled synchronously by the amplified electronic signal. Secondly, the analog signal received by photodetector is collected and converted to digital signal. Thirdly, the digital signal is separated into n groups with the central wavelengths of 1 , 2 : : : n , respectively, and then the data of n groups are synchronously processed in the FPGAR with hardware Fast Fourier 1 Q Transform (FFT). Through the integration of R.v/ D 1 R.v/ exp .2i v/ d , where  is the wavelength, and R(v) is FFT spectra with the peak frequency v of F1 , F2 : : : Fm , corresponding to different cavity lengths, the component of each frequency channels is filtered. R 1 Finally, the Inverse Fast Fourier Transform Q exp .2i v/ d v (Zhang et al. 2012b), is (IFFT), i.e., R ./ D .1=2/ 1 R.v/ performed for the m units simultaneously. As a result, all the sensor parameters

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Fig. 12 Schematic of the parallel processing flow for the TDM/WDM microstructure sensor network. (Copyright 2019 IEEE)

can be recovered with high speed. According to this mechanism, a demodulation system with high wavelength resolution of 3 pm and high speed of 500 Hz in the C band was realized, demonstrating the possibility of quickly retrieving the sensing information for the TDM/WDM microstructure sensor network (Cheng et al. 2018). Beneficial from the high spatial resolution and large capacity of the TDM/WDM microstructure sensor network , a high resolution manometry (HRM) was developed for measuring the pressure and motility of the gastrointestinal tract (Samo et al. 2016), by inserting the packaged sensing fiber into the gastrointestinal tract. Figure 13 illustrates the photographs of the HRM device with gastrointestinal tract pressure monitoring, packaged TDM/WDM microstructure sensing fiber with spatial resolution less than 1 cm, and the schematic of the HRM testing. Because the bare fiber is only sensitive to the axial strain but almost insensitive to the lateral strain, pressure transducer is necessary for the sensing fiber to enhance the pressure sensitivity. As displayed in Fig. 14a, biocompatible silicon rubber

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Fig. 13 HRM system based on TDM/WDM microstructure sensor network

was utilized to package the sensing fiber as a pressure transducer and protective coating. Regarding the linear deformation model of the silicon rubber under small pressure, the effective elastic modulus Eeff of the composite structure can be calculated as: Eeff D

Af Ef C Ap Ep Af C Ap

(4)

where Af , Ap are the cross-sectional area of the fiber and the polymer, and Ef , Ep are the elastic modulus of the fiber and the polymer, respectively. When the fiber was encapsulated in the center of the polymer with the diameter of 3 mm, the pressure sensitivity of the sensor was elevated to 2.22 nm/Mpa, which is much more sensitive than the bare FBGs sensor of 3 pm/Mpa. Further, the dynamic pressure response was investigated by tracking the pressure waves along the fiber. Figure 14b illustrates the experimental setup, where a 2 cm/s pressure wave was simulated by rolling a 100 g cylindrical metal stick over the packaged fiber, equivalent to the gastrointestinal tract pressure in vivo. The map of the real-time response is depicted in Fig. 14c, achieving the measurement of velocity, orientation, and value of the pressure wave, which provides information for the diagnosing clinician to find out the motility of the gastrointestinal tract clearly (Zhang et al. 2019 to be published).

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Fig. 14 (a) The photo of the packaged sensing fiber; (b) experimental set-up for pressure wave measurement; (c) the map of the real-time pressure response

Quasi-Distributed Sensor Network Based on 3D Encoded Microstructures Based on the low-finesse FPI composed of two closely spaced UWFBGs, WDM/FDM/TDM quasi-distributed sensor network is further investigated to realize huge multiplexing capacity in a single fiber. On account of the FPI structure and weak reflectivity, the microstructure can be 3D encoded with different wavelength, frequency, and time slot, named as Wi, fj, and Tk, which are realized by choosing different central wavelengths B of UWFBGs, different spatial distances between the UWFBGs pairs (defined as the cavity length LC ), and delay fiber with certain length. The configuration of the WDM/FDM/TDM quasi-distributed sensor network is described in Fig. 15, including the microstructured optical fiber and the central office for demodulation. When a probe light pulse is launched into the sensing fiber, microstructures with same time code are first located and distinguished through the time delay of received pulses roughly. The demodulation module analyzing the spectrum to obtain the wavelength and frequency information can be used to locate every single microstructure. Owing to this 3D encoding mechanisms,

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Fig. 15 Configuration of the WDM/FDM/TDM quasi-distributed sensor network: (a) experimental setup; (b) driving signals of AOM, TFF, and ADC in M-OTDR. (Copyright 2019 Springer)

microstructures spaced closely to each other can be further separated through wavelength and frequency encoding within one time domain, overcoming the restriction of spatial resolution in TDM scheme. The multiplexing capacity could reach to 18,000 when the reflectivity of the UWFBG is as lower as 40 dB (Sun et al. 2017). It can be seen that the quasi-distributed sensor network based on 3D encoded microstructures is able to provide enough sensing points and very flexible configuration for different kind of sensing applications. As shown in Fig. 15a, the demodulation scheme is based on a tunable FP filter (TFF) and 3D decode through spectrum analysis. The probe light from Amplified spontaneous Emission (ASE) source is modulated into pulses by the acoustic optical modulator (AOM). The modulated light pulses are directed into the sensing fiber. The back-scattered pulses carried with sensing parameters are amplified by an erbium-doped optical fiber amplifier (EDFA) and filtered by the TFF. Avalanche photo detector (APD) transfers the optical signals into electrical ones. The DAC&ADC module (composed of an NI 5781 adapter module and an NI 7962 FPGA module) synchronously controls the modulate time of the AOM and the TFF through DAC, as well as the sequential logic of electrical data captured by ADC. The driving voltage of AOM is a series of voltage pulses with the width of 200 ns, while that of the TFF is modulated in a sawtooth wave (Wang et al. 2018a). It should be noted that scanning nonlinearity and temperature sensitivity of the TFF will seriously affect the demodulation accuracy and stability. To resolve this issue, an improved demodulation scheme with self-calibration to actively compensate the error induced by TFF was proposed, as illustrated in Fig. 16a. A wavelength calibration unit with multiple reference FBGs is utilized with 3-order polynomial fitting to auto-calibrate the real-time relationship of TFF, and thus to eliminate the demodulation error. Note that the reference FBGs, which can be replaced with any optical filter device, are placed in an incubator chamber to keep the wavelengths constant. Pre-scanning and Polynomial fitting of the TFF function

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Fig. 16 Schematics of the demodulation platform with self-calibration: (a) system configuration; (b) the RMSE at different Polynomial fitting orders from 1 to 9; (c) spectrum of reference FBG s and the fitting curve. (Copyright 2019 IEEE)

is carried out by launching the tunable laser. Fittings with different fitting orders are established, and the root-mean-square errors (RMSE) are depicted in Fig. 16b, which indicates that linear fitting will induce greater demodulation error and 3-order polynomial fitting is enough for the demodulation. The detected spectrum of FBGs at a random moment and its real-time fitting curve are depicted in Fig. 16c, compensating the demodulation deviation (Wang et al. 2018a). The experimental results demonstrated that the demodulated wavelength deviation was only 6 pm when the temperature of TFF changes for 9.3ı C, the wavelength demodulation resolution was 1 pm, and the long-term demodulation precision was 3 pm, which could provide reliable measurements in large engineering projects.

Special Fiber Grating Sensor Network and Applications In comparison with ultraviolet (UV) lasers, femtosecond lasers may be a more powerful and versatile inscription source and have opened up a new territory for grating and microstructured fiber devices. Apart from seeing many successful commercial deployments of fiber grating sensors, novel and function-enhanced

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grating-based sensor structures have been developed for unique and high function sensors, by utilizing specially modified or tailored grating structures and special fibers. The exploitation of special grating structures and fibers and the combination with micro-nano features have broadened the sensor technology field for novel and high function optical fiber sensors. In the following, several special fiber grating sensors for network and their applications will be studied.

Tilted Fiber Grating (TFG) Sensor Network and Applications for near Infrared Detection (NID) In some applications, such as food safety, petrochemical process control, and pharmaceutical production quality control and agriculture area, the indirect detection method was not suitable any longer. So far, the main detection method used in those areas was analytical chemistry technique, which included near infrared spectroscopy, gas/fluid chromatography, nuclear magnetic resonance etc. Among them, near infrared (NIR) technology owns many advantages such as rapid process, nondestructive, noninvasive, chemical free, universal application, and suitable for process analytical technology and quality control (Woodcock et al. 2008; Jamrógiewicz 2012; Cleve et al. 2000). In NIR detection field, silica optical fiber has been employed as light energy transmitting optical fiber for application in harsh environment, owing to its low transmission loss at the NIR bandwidth. However, it has always been the focus of scientific research and exploration how to use the fiber to realize the distributed multi-point NID. There are several techniques to lead light transmitted inside the fiber core to out of fiber, including side-polished fiber, taped fiber, and fiber grating. Using side-polished fiber and taped fiber, light could interact with analyte by evanescence wave, which has very limited detection depth. The radiation of fiber grating offers a power controllable and effective method to achieve the analyte detection. Specifically, the 45ı TFG is the most effective radiation fiber grating, by which the light transmitting inside the fiber core could be partially coupled out. Figure 17a presents the working principle of the 45ı -TFG based on Brewster law, in which the light of transverse electric (TE) polarization is coupled out of fiber core and into radiation modes, and the light of transverse magnetic (TM) polarization still transmits inside the fiber core. Hence, the 45ı -TFG can be treated as an ideal in-fiber power taping device. According to the previous analysis (Yan et al. 2011, 2013), the taping ratio of TE polarization depends on UV-induced index modulation and the length of grating, which are easily adjusted by controlling the exposing time and grating length. Figure 17b shows the simulated results of taping ratio with different index modulations and grating lengths at the wavelength of 1500 nm. The most organic molecules have their fundamental characteristic absorption band at infrared area; their combine band would be located at the NIR area. In the mathematic, the light propagation in absorbing materials can be described using a complex valued refractive index. The real part of the refractive index indicates the phase velocity, while the imaginary part indicates the amount of absorption

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Fig. 17 (a) The structure diagram of 45ı -TFG; (b) taping ratio of TE polarization of 45ı -TFG with different grating lengths

loss when the electromagnetic wave propagates through the material. In terms of the absorption mechanism, the 45ı -TFGs-based NIR detecting system consisting of 45ı -TFGs, low-cost photodetectors, and single-wavelength laser diodes would be potentially used for distributed NIR detecting system. As shown in Fig. 18, the developed prototype 45ı -TFG NIR detection system is an optical fiber transmission system, which can be used for multi-point detection. Each detection unit contains one major probe grating and one reference grating. During the measuring process, the light coupled from the reference 45ı -TFG is directly reflected to the detector without any interaction with sample. While the light coupled from the probing 45ı -TFG is launched to the surface of sample, the reflected/scattered light signal will give information on the probe content due to absorption. A range of flour samples with different moisture levels were subjected to the measurement under this system. The different moisture level flour samples were prepared by leaving the fully dried (0% moisture level) flour sample in the air for

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Fig. 18 Configuration of 45ı -TFG-based distributed NIR detecting system

a period, then by weighting the mass increment of sample, the moisture level of flour sample was calculated. Fully dried flour sample was prepared by keeping the flour sample at 80 ı C oven for 24 h. To make sure the testing flour sample has uniform density and maintains at certain moisture level, in the experiment, 1g weight dried flour sample was filled into small container and left in air to absorb water and then sealed into a small container. Figure 19a shows NIR absorption spectra of flour sample with 10% moisture and 11.4% protein content measured from 1260 to 1580 nm, indicating a similar shape with the one shown in the reported research (Manley 2014). The flour samples were also tested with different moisture levels from 0% to 15.6% at 1450 nm (see in Fig. 19b). The result shows Log(1/R) Ireflectance (R D aI , where Ireflectance is the signal intensity reflected from the surface of reference sample; Ireference is the signal intensity of reference light; a is the correction factor.) is linearly proportion to the moisture level with high sensitivity around 0.014/% and low mean square error around 0.977. Figure 20 displays the first 45ı -TFG-based NIR detection prototype system, which has been installed in the UK Warburtons baking production line for flour moisture test. The benefits of this 45ı -TFGs-based NIR detecting system are low loss, compact structure, collimation free, and suitable for long distance, distributed operation. In future, the NIR system would be optimized to compensate the measuring error caused by physical property of sample, such as particle size, and achieve multicomponent measurement.

Er3C Doped Fiber Grating Sensor Network and Applications Compared to the passive Bragg gratings, fiber grating lasers, which are realized by inscribing gratings in Er3C doped fiber (EDF), not only possess a higher SNR but also offer a large-scale multiplexing capability. Actually, the fiber grating laser sensors mainly adopt the distributed feedback fiber laser (DFBFL) and the

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Fig. 19 (a) The NIR absorption spectra of flour sample with 10% moisture level and 11.4 protein content; (b) the measurement for flour sample with different moisture level at 1450 nm

distributed Bragg reflector fiber laser (DBRFL) as the sensing elements, which can be categorized into wavelength encode sensor and polarizer encode sensor with respect to their detection mechanism. Specifically, the DBRFL operates in single longitude mode with two orthogonal polarization modes and converts the measurand into change in the polarization modes beat frequency. Thanks to the advantages of easy interrogation, high measurement resolution, absolute encoding, and so on, the DBRFL sensors have attracted considerable interest in recent years.

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Fig. 20 Flour moisture detection through a compact 45ı -TFG-based distributed NIR detecting system

As illustrated in Fig. 21a, the DBRFL sensor is composed of two wavelengths matched Bragg gratings pair written in a short length of EDF directly. Due to the fiber birefringence, the laser always operates in two orthogonal polarization states, and when the laser output is monitored by a photodetector (PD), the two polarization modes will generate a beat signal in the radio frequency domain (Wo et al. 2012). The beat frequency is given as (Guo et al. 2011):  D Bc=neff 0

(5)

where c is the light speed in vacuum, 0 is the laser wavelength, neff and B are the average refractive index and birefringence of the optical fiber, respectively. By carefully optimizing the cavity parameters such as the absorption of EDF, reflectivity, Bragg wavelength and bandwidth of the gratings, single-frequency lasing can be easily achieved when the cavity length reduces to several centimeters (see Fig. 21b). Meanwhile, very low noise floor of the beat signal can be achieved, where the noise power spectral density exhibits a C / f profile with the factor C about 3.923  104 (Hz2 ) (as shown in Fig. 21c). When an external measurand is applied on the laser cavity, the fiber birefringence will change linearly due to the elasto-optic effect, as well as the beat frequency of the two polarization modes. Similarly to the passive FBG sensors, the DBRFL sensors can be multiplexed in a single fiber through WDM and FDM . A sensor array consisting of as many as 16 DBRFLs have been presented, which were wavelength multiplexed by inscribing fiber gratings with different pitches, and frequency multiplexed in RF domain by controlling the intracavity birefringence. The lasing wavelengths range from 1528.8

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Fig. 21 Schematic diagram of DBRFL: (a) structure and principle of DBRFL; (b) photo of the short cavity DBRFL; (c) frequency noise power spectral density of the DBRFL

Fig. 22 The optical spectrum (a) and frequency noise spectra (b) of the multiplexed DBRFL sensor network. (Copyright 2019 OSA)

to 1566.4 nm with the spacing of 2.4 nm and SNR higher than 35 dB, and the output beat frequencies ranges from 140 MHz to 1.7 GHz, controlled by the CO2 laser side irradiation to the laser cavity (Jin et al. 2014). Besides, through specially design the DBRFLs to eliminate out of band reflections (minimize laser-laser interactions), decrease the laser pump threshold (1 mW at 1480 nm) and energy absorption per device ( 1. According to the size of the object to be trapped, the model will be discussed in the Rayleigh regime (Ashkin et al. 1986) and the Mie regime (Ashkin 1992), respectively. In the Rayleigh regime with an object size of 2r < , the scattering force can be determined by Ashkin et al. (1986) Fs D

2  I0 128 5 r 6 n2  1 nb P s D nb : c c 34 n2 C 1

(1)

Here, Ps is the scattered power, I0 is the incident intensity, and n and nb are the effective index and the index of the surrounding medium, respectively. With effective refractive index of n > 1, Fg attracts objects to the location with higher laser intensity, often near the focus. The gradient force for a spherical Rayleigh object with a polarizability of ’ can be expressed as Fg D 

n3 r 3 nb ˛rE 2 D  b 2 2



 n2  1 rE 2 n2 C 1

(2)

The optical force is generated due to the momentum transfer from photons to the microparticle. When the diameter of objects 2r  , the momentum transformation

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Fig. 1 Description of optical force on a dielectric particle

process can be simply described with ray optics in Mie regime (Ashkin 1992). Three types of beams, i.e., divergent beam, parallel beam, and convergent beam, are described, respectively, in Fig. 1a–c. I1 and I2 denote two symmetrical rays. The light-matter interaction occurs on the surface of microparticle, where the momentum transfers from photons to microparticle, inducing two relevant optical forces, F1 and F2 . Ftotal denotes the total optical force induced by the laser beam and its direction is also given. In Fig. 1a, b, the direction of Ftotal is the same as the laser beam, resulting in a pushing force on the microparticle. Ftotal shows a reverse direction when the beam is convergent. In Fig. 1c, the microparticle will be pulled back and keep balanced near the focus of laser beam. For microparticles with relatively larger diameter, optical force can be expressed in the Mie regime (Ashkin 1992). Ftotal can be broken into two components, FZ and FY , which are determined by Fs D

nP c

 1 C R cos 2 

nP Fg D c



T 2 Œcos .2  2 / C R cos 2 1 C R2 C 2R cos 2

T 2 Œsin .2  2 / C R sin 2 R sin 2  1 C R2 C 2R cos 2

 (3)

 (4)

P is the power of the laser beam.  is the angles of incidence and refraction. R and T are the Fresnel reflection and transmission coefficients. Fs is along with the optical axis and is regarded as the scattering force to push the microparticle away from the source. Fg is perpendicular to the beam and can be considered as the gradient force which can attract the microparticle close to the optical axis.

Optical Fiber Tweezers Conventional optical tweezers employ the microscope objective with a high numerical aperture (NA) (Ashkin et al. 1986; Ashkin 1992). The bulky device make optical tweezers difficult to use and expensive. Optical fiber tweezers (OFT) offer advantages over conventional objective of low cost, flexibility, long transparent distance, and easy integration. However, it is a big challenge to enhance the trapping efficiency for OFT due to the low NA of the fiber. Three main categories, i.e., lensed

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Fig. 2 OFT based on lensed fiber

fiber or fiber taper, dual-fiber, and special constructions, have been developed to resolve this problem. Fiber taper is a straightforward way to enhance the efficiency of 3D trapping (Fig. 2). The tapered structure strongly focuses the laser beam so that the stable 3D trapping can be achieved (Liu et al. 2006). The fiber tapers can be fabricated by polishing (Kostovski et al. 2014), chemical etching (Gong et al. 2013, 2014; Chuang et al. 1998), or the heating-and-drawing methods (Liu et al. 2006). These methods are easy to fabricate and simple to operate. However, the sharp geometry corresponds to a short focus length and limits the operation range by 3D trapping the microparticle very close to the fiber end. For resolving this problem, OFTs based on dual-fiber are proposed. The dualfiber optical tweezers makes use of two aligned optical fibers with flat or lensed tips so that the laser beams emerging from the fibers are counter-propagating along a common optical axis (Guck et al. 2000). In this case, the trapping distance can be lengthened due to the balanced axial scattering forces (Jess et al. 2006). Since the fiber alignment is crucial for this arrangement, the two fibers are generally embedded in a substrate to facilitate the alignment, which limits the flexibility of OFTs. Moreover, due to the relatively large size of the optical fiber, it cannot trap particles at sub-wavelength scale or particles lying on the substrate. In 2000, an inclined dual-fiber optical tweezers were proposed by Taguchi et al. (2000), which can achieve levitation of a microscale object. It has better flexibility than the counterpropagating dual-fiber optical tweezers. Accompanied with the development of fabrication technologies, OFTs with better performance based on special structures are proposed. Cristiani et al. proposed an OFTs based on a multi-core optical fiber (Liberale et al. 2007). The multi-cores were shaped with proper angles so that laser beams from these cores were reflected into a tight focus. Strong gradient force enables 3D trapping near the focus. It can

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trap and manipulate microparticle over a relative long distance with flexibility better than the dual-fiber optical tweezers. However, the complex manufacturing process hinders its broad applications in real world. In 2009, Yu et al. developed an inclined dual-fiber optical tweezers, by which both manipulation and force sensing (Liu and Yu 2009) were achieved. Different from the traditional dual-fiber optical tweezers, there was an inclination angle ( ) between two fibers. With the help of the optical force applied by the two optical fibers, the object can be trapped below the intersection of the two beams. The inclined dual-fiber optical tweezers extended the distance of the equilibrium position, so it can trap larger particles of tens of micrometers without physical contact. The trapping efficiency changed linearly with the displacement in the range of 1 m to C1 m. Force sensing was demonstrated with the help of a positionsensitive photodiode, which was used for detecting the position of the trapped beads. By detecting the displacement, the external forces applied on the particle can be measured. The  is an important parameter for this device. When   45ı , the device can trap the bead in x and y axes, but cannot lift it. If   50ı , it can achieve trapping in three dimensions including lifting. With larger  , the OFT can achieve optical force strong enough in z axis for lifting. The inclined dual-fiber optical tweezers is more compact than the counter-propagation OFTs due to the less limitation on the fiber alignment.

Dual-Beam Optical Manipulation The OFT can trap an object, but cannot control or adjust its position without moving the OFT. This weakness limits the functions and applications of OFTs. The dualbeam fiber trap (DFT) was first proposed to solve the problem. Comparing with the conventional structure based on two focused laser beams, the dual-beam optical manipulation has the advantages of easy fabrication and easy integration, which makes it suitable for interdisciplinary research. More importantly, the unfocused laser beams avoids thermal damage to the trapped object, making it suitable for biological use. The DFT is composed by two counter-propagating laser beams with Gaussian intensity profile (Ashkin 1970). If the refractive index of the object is larger than the surrounding medium, it can be stabilized within the laser beams. One special advantage of DFT is that the trapped object can be stably and flexibly controlled between the two fiber ends. The schematic principle of DFT is shown in Fig. 3. Two counter-propagating beams from optical fibers formed a dual-beam trap. With the help of microfluid, objects can be loaded automatically and continuously, which could improve the throughput of the device. The object can be pushed and localized to different positions by controlling the laser power, P1 and P2 . The DFT can trap one or several objects and move them to any position on the axis between two fibers. Different numbers of yeast cells were trapped, as shown in Fig. 4, named as optical binding. For objects with ellipsoidal structure, they were often assembled along with their macro-axis. By controlling the power from each

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Fig. 3 Schematic principle for dual-beam optical fiber manipulation

Fig. 4 Manipulation of different numbers of objects based on DFT

Fig. 5 Manipulation of a cell string to different positions based on DFT

fiber, the position of the trapped cell string can also be adjusted. The objects can be controlled to the right when the laser power from the left fiber is larger than the right (Fig. 5a). Increasing the power from the right fiber, the cell string can be gradually pushed to the left. When the power from the two fibers is equal, the objects were trapped in the middle (Fig. 5b). With increasing power from the right, the objects were pushed to the left (Fig. 5c). In cytology, the Raman spectroscopy is very useful for cell identification by detecting the chemical composition. The function of Raman spectroscopy can be further enhanced by DFT. In 2011, scientists achieved tumor cell identification based on the combination of Raman spectroscopy and DFT (Guck et al. 2001). Two 1070 nm lasers were coupled into two single-mode fibers (SMFs) to construct a DFT and used for trapping the cell. With the help of microfluidics, the device can automatically distinguish the cell types including RBCs, leukocytes, acute myeloid leukemia cells, and breast tumor cells. Besides optical trapping, the DFT can also deform the living cell, if the laser power is strong enough. Guck and co-workers reported a unique DFT, named optical stretcher (Dochow et al. 2011). They trapped single cell between two opposite,

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unfocused beams from SMF. Although the total force on the object is zero, the forces on the cytomembrane can be as high as several hundreds of pN. The trapped living cell can be stretched along the optical axis. Besides, due to the unfocused laser beams, the optical stretcher with a laser power as high as 1.7 W did not damage the living cell. It can be used to measure the cell elasticity, further to distinguish cancer cells from normal ones. Based on the strong stretching force, it can also be used for detecting the cytoskeleton quantitatively. As it is easy to be integrated with microfluidics, the optical stretcher has great potential for high throughput detection of the cell elasticity. The DFT can also be used for temperature sensing (Zhang et al. 2014). They used etched fibers to form a more stable optical trap. The microparticle was trapped in a sealed cell that was fabricated by packaging a dual-fiber optical manipulation system in a quartz capillary. The fiber ends were chemically etched to a concave shape. For sensing application, they adjusted the position of the microparticle by tuning the power of 980 nm laser, and the shift of interference spectra near 1550 nm was used as sensing signals. Hollow capillaries have the potential for guiding both light and liquid in it. However, high transmission losses limit the guiding distance. Hollow-core photonic crystal fiber (HC-PCF) is an excellent candidate for both optical waveguiding and liquid sampling. Russell and his co-workers introduced a tool for single-cell manipulation based on the liquid-filled HC-PCF (Unterkofler et al. 2013). First the cell was trapped in front of the fiber core with a focused laser beam vertically to the fiber. Then, a horizontal beam irradiated on the cell and pushed it into the fiber core. The cell was transferred along the HC-PCF in the core with a long distance of tens of centimeters. Besides, cell deformation was also observed during the optical manipulation due to shear force, making it promising for biomechanical detection of living cells. DFT based on HC-PCF is also an effective tool for sensing. Detection of multiple parameters with high spatial resolution was realized based on HC-PCF DFT (Bykov et al. 2015). A microparticle was trapped in the hollow core of HC-PCF, and the position was adjusted by the counter-propagating laser. Parameters of the environment around the fiber, including transverse mechanical vibration, temperature, and electric or magnetic field, can be detected by monitoring the reflected or back-scattered light.

Single-Beam Optical Manipulation The single-beam optical manipulation (SOM) could improve the flexibility and integration, compared with the dual-beam optical manipulation. The SOM employs a single laser beam from an optical fiber, so it can be easier to move to the object to be manipulated, compact for integration with microfluidics, and has wide manipulation area. It might achieve internal detection of tiny object like cells by inserting a tapered fiber into it.

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Fig. 6 The principle of the SOM based on a flat-facet SMF

Principles The principle of the SOM is schematically shown in Fig. 6. The optical manipulation along with the optical axis is based on the force balance on the microparticle between the axial optical force, Fao , and the microfluidic flow force, Fv . As the flow rate v increases, the flow force increases according to the Stokes law: F v D k1 

(5)

Here k1 D 6 a, with the coefficient of viscosity of water and a the radius of the microparticle. Therefore, the Fv tends to push the microparticle toward the fiber facet. On the other hand, the Fao increases as the manipulation length, Lm , decreases, which forms a counterforce to push the microparticle away from the fiber end. The manipulation length, Lm , is defined by the distance from the center of the microparticle to the fiber tip. The total force on the microparticle can be balanced at certain Lm , which uniquely corresponds to the flow rate. The force balance can be expressed as Fv .v/ D Fao .Lm /

(6)

Thus the Lm , can be determined by both the optical laser power and the flow rate. The gradient force points to the direction of the light intensity enhancement and can be divided into two components, the transverse gradient force, Ftg , and the axial gradient force, Fag . Ftg compensates the difference between the gravitational force and the buoyancy force and confines the microparticle in the optical axis. In most cases, the density difference between the microparticle and the liquid is small so that the weak gradient force is more than enough to trap the microparticle on the axis. The Fao consists of the scattering force, Fas , and the axial gradient force, Fag . The Fag is very small and is negligible compared with Fas and Fv and can be

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neglected in the axial direction. Therefore, the force microbalance can be rewritten approximately as Fv .v/ D Fas .Lm /

(7)

The scattering force is proportional to the laser intensity (Harada and Asakura 1996), i.e., Fas D k2 I

(8)

Here k2 is a constant related to the particle radius, the refractive index of the particle, and the environmental medium and the laser wavelength. I is the intensity distribution of the laser beam. The output from the fiber can be considered as a Gaussian beam, and the output intensity is determined by  I .r; z/ D I0

!0 !.z/

2 e



2r 2 ! 2 .z/

(9)

q Here I0 is a constant intensity, and !.z/ D !0 1 C .z=zR /2 is the beam radius at the axial position of z, with zR D ! 20 =n0 . ! 0 is the waist radius of the beam, and n0 is the refractive index of the solution. Considering the microparticle on the z axis, i.e., r D 0, the intensity can be expressed as a function of the manipulation length, Lm : i h I .0; Lm / / 1= 1 C .Lm =zR /2

(10)

Considering the Eqs. 5, 7, 8, and 10, the relationship between the flow rate and the manipulation length can be described as h i  / 1= 1 C .Lm =zR /2

(11)

zR is about 20 m in the experiment. With a relatively large manipulation length, i.e., (Lm /zR )2  1, the relationship between the flow rate and the manipulation length can be simplified to be  / (Lm /zR )2 . In the log-log scale, the Lm  v curve is approximately linear and the slope is k D log Lm = log v D 0:5

(12)

Graded-Index Fiber Taper Optical tweezers are mainly developed as a tool for 3D trapping the object. However, it suffers from the limited manipulation length (Lm ), inherently owing to the small core diameter of the SMF and small radius of curvature of the fiber tip. The manipulation distance can be defined by the fiber end surface and the stable trapping

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point of the microparticle. It makes this kind of optical tweezers inflexible to be used, just like a short-working-distance objective in an optical microscope. It might even damage the biological samples due to a short Lm of the OFTs. Therefore, OFTs with a long Lm is meaningful for realizing a truly non-contact optical manipulation. In 2014, Gong et al. proposed a single-fiber optical manipulation with a long adjustable distance of more than 40 m by introducing the periodic focusing effect of graded-index fiber (GIF) into OFTs (Gong et al. 2014). As shown in Fig. 7, a fiber-coupled laser of 980 nm is employed as the light source for the optical manipulation. One percent of the light is used to monitor the laser power. The light transmitted from the 99% port of the coupler is coupled into the graded-index fiber (GIF), which is with a core diameter of 62.5 m and an NA of 0.27. The single-mode fiber (SMF) and GIF are aligned by a silica capillary. An air cavity, introduced between the lead-in SMF and the GIF, can be precisely adjusted by a three-dimensional (3D) translation stage fixed with the lead-in SMF. The light distribution in the GIF can be optimized by adjusting the air cavity length. The light beam emerge from the GIF taper, which was fabricated by the process of the socalled two-step etching technique (Chuang et al. 1998). The microscopic image of the fabricated GIF taper is shown in the insert of Fig. 7, whose cone angle was measured to be about ’ D 58ı . The performance of optical manipulation is improved mainly by two factors. One is by introducing the focusing effect of GIF taper. The other is by introducing an air gap with tunable cavity length (Lair ) between the lead-in SMF and the GIF taper. The GIF has an index profile with quasi-parabolic function in the radial direction

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Fig. 7 Experimental setup for single-fiber optical manipulation by GIF taper

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Fig. 8 (a) Lm versus time when increasing the laser power from 15.0 mW to 45.0 mW and (b) Lf versus the air cavity length at different GIF length

of the fiber core, which leads to periodic convergence and divergence of the light beam propagating in the GIF. So it has better focusing effect than SMF. Besides, it is fabricated with a large cone angle of 58ı which is very helpful to obtain a large gradient force and to form a stable optical trap (Mohanty et al. 2008). With the help of the air cavity, stable optical trapping of yeast cells is observed in aqueous solution with a constant flow rate. It is also demonstrated that the manipulation distance can also be extended by changing the pump laser power. Figure 8 shows the optical manipulation of a microparticle that was influenced by the laser power (Fig. 8a) and the air cavity length (Fig. 8b). The Lm versus time when increasing the power from 15.0 mW to 45.0 mW was shown in Fig. 8a. The cavity length is 100 m, and the flow rate is 12 m/s corresponding to a constant viscous force of 0.54 pN. At t D 5s, Lm was increased gradually from 17.9 m once the laser power was increased. At t D 20s, while Lm was 47.4 m and tended to approach a stable state, another yeast cell was trapped together with the first one and Lm further increased to 61.5 m and then remained stable. The air cavity length can also manipulate the microparticle, and the results were reflected by Lf defined as the distance between the fiber tip and the focus point with the maximum intensity. With a GIF length (LGIF ) between 650 m and 850 m, one can see that the Lf decreases with Lair . With LGIF between 900 m and 1050 m, the evolution of the Lf did not change monotonously with Lair , as Lf changes from a negative value to the positive. It is because in this range, Leff D LGIF C Lair changes from smaller than to larger than two times of the GIF pitch. With the GIF length changing from 650 m toward 1050 m, the Lf  Lair curve gradually shifts to the left, with a smaller Lf . For LGIF D 1150 m, the Lf  Lair curve coincides with that of LGIF D 650 m. This indicates that increasing the air gap is equivalent to increase the GIF length. However, adjusting the Lair is much easier than precisely controlling the GIF length. The manipulation length, Lm can be controlled by changing the balance between the optical force and the microfluidic flow force, as shown in Fig. 9 (Gong et al. 2015a). When fixing the laser power and the microcavity length during this

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Fig. 9 Manipulation length versus flow rate at different laser powers

Fig. 10 Microscopic images of (a) a sharp GIF taper and optofluidic tunable manipulation with Lm of (b) 36.2 m and (c) 177.0 m

experiment, the light distribution, thus the optical force distribution, for trapping certain object were also fixed. Lm of the microparticle was tuned by the flow rate for breaking and reconstructing the force balance. This mechanism demonstrated the controllable optical manipulation and can also be considered as a prototype for flow rate sensing. Note that an abruptly fall of manipulation length from 20 m (hollow circles in Fig. 9) to 3.5 m (in contact with the fiber tip) happened at all five laser powers. The last balanced position was independent with laser power and flow rate. Thus it is reasonable to consider it as the focus of light emitting from the GIF taper and is determined only by the light distribution. By using a GIF taper with a special shape, the range of Lm can be tuned from 36.2 m to 177 m (Fig. 10). Different from previous GIF tweezers fabricated by self-terminated chemical etching method (Gong et al. 2013, 2014; Chuang et al. 1998), this GIF taper was fabricated by heating-and-drawing method. One advantage is that the loss near the GIF taper is lower so that the gradient force is strong enough to trap the object on the optical axis, even at long manipulation length.

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By introducing an air microcavity and adjusting its length, Lair , the periodic focusing effect of light in the GIF taper and therefore, Lm , can be adjusted. In this case, the microparticle can be balanced at the focus of the GIF taper by increasing the flow rate slowly. When the force is balanced at the focus, Lm can be controlled by tuning Lair (Fig. 11). During the tuning, the laser power and the flow rate were fixed at 43.5 mW and 450 nL/min, respectively.

Flat GIF Previous optical manipulation were often based on fiber tapers, until Gong and his co-workers developed a new method based on GIF with a flat endface. The fabrication process is very simple by cleaving the GIF for optical manipulation, compared with tapering the fiber. Thanks to the quasi-parabolic index profile of GIF, a strain-controllable optofluidic manipulation technology based on GIF was developed. Lm of up to 1314.1 m was demonstrated (Zhang et al. 2016). The GIF-based controllable optical manipulation is achieved by directly stretching the GIF to change the light distribution, due to periodic convergence and divergence of the light beam propagating along axial direction of GIF. The periodic manipulation of microparticle is shown in Fig. 12. Figure 12a shows a sequence

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Fig. 12 (a) Microscopic images of optofluidic controllable manipulation of the single microsphere with strains of 0, 95 ", 286 ", 571 ", 667 ", 1048 ", 1143 ", and 1238 ", respectively, and (b) manipulation length, Lm , versus strain at flow rates of v D 150 nL/min

of the microscopic images with different strain on the GIF with a fixed power of 149.29 mW and a flow rate of 150 nL/min. Lm varied with the strain with a clear period around 953 " corresponding to a GIF length change of about 500 m, which is in agreement with that of the GIF pitch. The flow rate sensing was demonstrated by detecting the Lm with fixed strain and laser power. Figure 13a shows the Lm as a function of the flow rate in the log-log scale. The flow rate can be detected between 30 nL/min and 3 L/min.

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Fig. 13 (a) The result about flow rate sensor with 1048 " and (b) selected microscopic images of green curve in (a) with flow rates of 30 nL/min, 40 nL/min, 50 nL/min, 80 nL/min, 200 nL/min, and 800 nL/min, respectively

Corresponding microscopic images of the optofluidic manipulation with different flow rate are shown in Fig. 13b. The laser power is fixed at 25.38 mW and the strain on the GIF is 1048 ".

Flat SMF The fiber taper can focus the beam and form a 3D optical trap. However, it is difficult to fabricate fiber tapers with high repeatability. It is also difficult to precisely control

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the length of GIF in order to obtain the desired light distribution near the flat GIF tip. It is the simplest scheme to use a single flat SMF for optical manipulation. All that is needed is cleaving the SMF to form a flat fiber tip. Combined with the flow force provided by the microfluid, the optical manipulation based on a flat SMF can be performed, which has the advantages of easy to fabricate and use, availability for mass production, and low cost. Thanks to these advantages, an optofluidic flow rate detection method was demonstrated (Gong et al. 2015b). A 980 nm fiber-coupled diode laser was used as the light source for the optofluidic manipulation. The absorption coefficient of both the water and PS microparticle is low at 980 nm. Therefore, the optical force dominates, while the photophoresis or negative photophoresis force can be neglected. The laser power was tunable for adjusting the optical force. One percent of the laser power was monitored for the calibration. Ninety-nine percent of the laser power was coupled directly into the cleaved SMF, which was inserted into the microfluidic channel for optical manipulation. The fiber was positioned against the flow. The microparticle with a diameter of 7 m was flew through the channel and trapped by the gradient force and also the axial force microbalance. The flow rate was controlled by a syringe pump. The manipulation length was measured by an optical microscope. According to Eq. 6 in section “Principles,” the flow rate can be determined by detecting the manipulation length. Note that the manipulation length is inversely proportional to the flow rate, that is, Lm increases as v decreases. Therefore, the optofluidic manipulation technique is particularly useful to detect low flow rate. The sensing performance of the device is shown in Fig. 14. The manipulation length as a function of the flow rate with different laser power is given in Fig. 14a. The maximum Lm was 715 m with a laser power of 146.3 mW and flow rate of 75 nL/min. The minimum Lm was about 3 m with a laser power of 11.4 mW and flow rate of 5000 nL/min. At a laser power of 41 mW, a dynamic range of about three orders of magnitude, from 20 nL/min to 14,000 nL/min, was achieved. The dynamic ranges for the flow rate detection are similar with different laser powers, while a relative lower (higher) flow rate can be detected with a lower (higher) laser power. In conclusion, with the coordination of the laser power and flow rate, this device achieved flow rate sensing with a large dynamic range from 20 nL/min to 22 L/min and manipulation range from 3 m to 715 m. The flow rate was controlled between 40 nL/min and 80 nL/min with a step of 10 nL/min (red solid circles, Fig. 14b) and between 100 nL/min and 300 nL/min with a step of 50 nL/min (blue solid circles). The manipulation length was recorded at each flow rate for 15s. The temporal stability of the trapped microparticle is good during the flow rate detection and can be sustained for several hours. The step changes of the manipulation length can be clearly distinguished. The sensing performance for flow rate was limited by Lm , which is one of the weaknesses of the flow rate sensor. The sensitivity became lower at shorter Lm . Although high sensitivity at long Lm , the optical trapping will be unstable due to the weak gradient force. To solve this problem, An idea of dual-mode detection was proposed (Gong et al. 2017).

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The dual-mode detection, i.e., open-loop mode and closed-loop mode, was developed as an optofluidic flowmeter with a large dynamic range of four orders of magnitude. The principle of open-loop mode is the same as that mentioned above. Lm is used as the sensing output, and the flowmeter has an inverse sensitivity so that it has higher sensitivity at lower flow rate. In the closed-loop mode, Lm is set to be

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Fig. 15 Schematic experimental setup for dual-mode fiber optofluidic flowmeter

constant, and a feedback signal, tuning the laser power for the force balance, is used as the sensing output. The principle of the closed-loop mode is shown in Fig. 15. Lm is monitored by the CCD and calculated by a homemade software in real time. In the closed-loop mode, the Lm is set to be a constant value, L0 . Once there was a slight deviation, i.e., L from L0 , a feedback signal was generated by tuning the laser power in order to keep the force balance on the microparticle and to keep the manipulation length constant at L0 . In this case, the laser power serves as the sensing output proportional to the flow rate. The closed-loop mode is helpful for extending the upper detection limit of flow rate and also enhancing the sensitivity at higher flow rate. The calibration of the optofluidic flowmeter was performed in both open-loop and closed-loop detection modes. In the open-loop mode, the laser power was firstly fixed at 23.5 mW, and the manipulation length was measured as a function of flow rate. The manipulation length Lm decreased as a function of the flow rate (Fig. 16a). At 23.5 mW, the limit of detection (LOD) was pushed down to 10 nL/min, and an upper limit of 5000 nL/min was obtained. The results indicated that lower LOD can be achieved with lower laser power, while the upper limit can be extended by using higher power. However, the gradient optical force was weak when using a small laser power, which may not be strong enough to trap the microparticle at a long manipulation length and the microparticle may flow away. And the flow rate may also have some fluctuations when the syringe pump was used to generate a very low flow rate. The stability of generating a low flow rate can be enhanced by using a low-volume syringe. However, a low-volume syringe was not helpful for detecting large flow rate as the solution in the syringe may run out in a short time. Although the upper limit of the flow rate might be extended by using higher laser power, the sensitivity was rather low at high flow rate range in the open-loop mode. This can be seen from Fig. 16b, by plotting the Lm in the linear scale. One can see

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Fig. 16 Calibration of the optofluidic flow rate sensor in open-loop mode with y axis in (a) log-log and (b) semi-log scale

that the Lm decreases very quickly and nonlinearly with an increasing flow rate. This is an inverse sensitivity that is perfect for getting ultrahigh sensitivity at the low flow rate range. But the decrease in sensitivity limits the dynamic range in the high flow rate range. Therefore, the open-loop mode is good for measuring low flow rate with high sensitivity; thus we chose the laser power of 23.5 mW as an optimized power for the dual-mode flowmeter. In the closed-loop mode, the compensated laser power was used as the sensing output and shown as a function of flow rate in Fig. 17. The Lm was first set at 60 m and achieved a dynamic range of 1000–20,000 nL/min. By reducing the constant Lm to 30 m and 15 m, the dynamic range was extended from 2000 nL/min to 50,000 nL/min and 5000 nL/min to 100,000 nL/min, respectively.

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Fig. 17 Calibration of the optofluidic flow rate sensor in the closed-loop mode with manipulation length fixed at 15 m, 30 m, and 60 m, respectively

Fig. 18 Sensing performance of the optofluidic flow rate sensor

The dynamic ranges by using the closed-loop and the open-loop modes are complementary with each other. Therefore, by using the dual-mode flowmeter, the dynamic range can be greatly extended. A threshold was set as 5000 nL/min, corresponds to a threshold manipulation length, Lm0 , and a threshold laser power, P0 , respectively. Initially, the laser power was set to be 23.5 mW for trapping a single microparticle, and the flowmeter was operated in the open-loop mode. If the initial flow rate was high so that the manipulation length was reduced down to Lm0 , the flowmeter was switched to the closed-loop mode, and the laser power was adjusted by the feedback signal in order to maintain the manipulation length to be 15 m. As shown in Fig. 18, the flowmeter achieved sensing with a large dynamic range of four orders of magnitude from 10 nL/min to 100,000 nL/min.

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OF2 Sensors Based on Photothermal Effect In this section, two sensing schemes based on heat transfer in a microring resonator and the generation of microbubble-on-tip will be introduced. The fiber optofluidic technologies based on photothermal effects by heating the microfluidic channel via fiber-coupled laser can be applied for sensing applications. The heat transfer in liquid-core microring resonator can be employed for flow rate detection. Microbubbles can be photothermally generated on the fiber tip that is coated with carbon nanotubes or gold nanofilm. This type of gas microbubble generated in microfluid can be employed as a micro-interferometer in which its free spectral range (FSR) could be measured for sensitive temperature and flow rate detection. By using optical imaging to monitor the evolution of the microbubbles on the fiber tip coated with gold nanofilm, concentration detection with a large dynamic range can be realized.

Heat Transfer in Microring Resonator Gong et al. reported an optofluidic flow rate sensor based on the heat transfer effect in a microfluidic channel for the lab-on-a-chip applications. Microscopic and SEM images of the flow rate sensor are shown in Fig. 19. The microfluid, whose geometrical structure is a hollow round capillary, acts as optofluidic ring resonator (OFRR). A fiber taper with a waist of about 3 m is perpendicular to the capillary for optical coupling and light collection. A small fraction of the incident light is coupled from the taper into the OFRR and is kept circulating in the wall of the OFRR due to the total internal reflection. After each round trip, a small fraction of light is coupled out to the fiber taper. The multiple output beams interfere with each other, and the resonant dips are generated in the spectrum of the output beam. In case of the high Q factor resonator, the full width at half magnitude (FWHM) of the linewidth is narrow and is helpful for the high-performance sensing.

Fig. 19 (a) Microscopic image of the structure for the flow rate detection and (b) the SEM image of the cross section of the OFRR. Bar: 50 m

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A 1480 nm pump laser was used for heating the microfluid due to the high optical absorption coefficient of water in this spectral range. The laser was coupled into a fiber tip, which was fabricated by heating and drawing a SMF. The fundamental mode of the microring was often used for sensing which was located near the outer surface of the capillary with larger radius and the effective index (neff ). The light intensity distribution, including the evanescent wave, of the fundamental radial mode will not penetrate into the inner liquid of the capillary. So the relation between the temperature and wavelength shift can be expressed as  

 wal l @neff

T: D ˛C  neff @nwal l

(13)

Here ˛ D 1/r(@r/@T) is the thermal expansion coefficient of the resonator, and  wall D @n/@T is the thermo-optic effect for the wall. The “wall” means the solid part of the capillary. Heat is at first transferred from the microfiber tip to the inner liquid core and then to the silica ring resonator. From Eq. 13, there are two factors, i.e., thermal expansion of the silica microring and thermo-optical effects, affecting the wavelength shift of the OFRR. The transmission spectrum at a flow rate of 10 L/min under different heating power is shown in Fig. 20. After tuning the laser power, the heat transfer got balanced in a few seconds, and then the spectrum was recorded. With the increase of heating power, the resonant dips show a right shift, indicating an increase in phase change in a round trip. The result agrees well with the theoretical prediction in Eq. 13. As shown in Fig. 21 in linear scale, the wavelength changes nonlinearly with the flow rate under different heating power. The sensitivity of flow rate sensor increases as the heating power increases. The maximum sensitivity of 57.6 pm per L/min was achieved at a heating power of 64.74 mW. Further increase in the heating power will led to unwanted microbubble generation. The data, replotted in log-log scale in Fig. 21b, show a good linearity.

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The flow rate detection with a dynamic range of 2–100 L/min was achieved (Fig. 22). With a heating power of 53.7 mW, the calibration curve of the wavelength shift versus the flow rate was measured and shown in the linear scale and the loglog scale in Fig. 22a, b, respectively. The error bar was calculated from the triplet measurements. The results indicate that the repeatability of the flow rate detection is good. The wavelength shift is set to be zero,  D 0, at the initial condition with no laser heating for each flow rate and corresponds to the reference wavelength of 0 . The data in Fig. 22a are fitted via  D a b . It is equivalent to the linear fitting of the data in Fig. 22b. The R square of linear fitting is about 0.993. The fitting curves are also shown in Fig. 22a, b as the black lines. The proposed sensor has the advantages of high flexibility and can perform local flow rate detection, especially for capillary-based microfluidics while not degrading other functionalities of the microring resonator including the biochemical sensing or optofluidic lasing.

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Microbubble-on-Tip Structure Laser-induced bubbles in microfluidics have been used as valves and pumps because they can be generated and cracked easily by a laser beam (Zhang et al. 2011). The generation of microbubbles is related to the properties of ambient environment, including temperature, viscosity, flow rate, and concentration, so that this kind of microbubble has the potential for sensing. In this section, two methods based on microbubble-on-tip (BoT) structure will be described. The first one can achieve temperature or flow rate sensing based on microbubble interferometer generated on a flat fiber tip coated with carbon nanotube (CNT) film. The evolution of the microbubble was monitored by detecting the changes of the reflective interference spectra. The second method is based on microbubbles generated on a flat fiber tip coated with gold film, which is used as a concentration sensor. The diameter of the microbubble is monitored by imaging.

Temperature and Flow Rate Sensor A fiber optofluidic interferometer for temperature and flow rate sensing in microfluid is developed based on the BoT structure (Zhang et al. 2017). The generation process and sensing mechanism of this BoT sensor are very different from traditional optical fiber sensors. Experimentally, two laser beams were delivered to the same fiber tip through a wavelength division multiplexer. A heating beam at 980 nm was employed to heat microfluid and generated the microbubble on the fiber tip that was coated with CNTs. The laser power was tunable from 0 to 300 mW. A tunable laser from an optical spectrum analyzer (OSA) (OPT162, 1505–1630 nm, Agilent) was used to monitor the changes of the reflective interference spectra that resulted from the evolution of the microbubble. The interference occurred between the Fresnel reflections from the solid/air surface and the other from the air/liquid surface. CNT film was optically deposited on the fiber endface to increase the laser absorption, because it has a low conductivity of 1.52 W/(mK) in the radial direction and also a low specific heat capacity of 0.7 J/(gK), which are helpful for increasing the absorption of pump laser and introducing a large temperature increment for the BoT generation. Therefore, the BoT structure can be generated with laser of 980 nm with a relative low power (Hepplestone et al. 2006). A microbubble was generated thanks to the increment of the gas content around the fiber tip, either from the separation of dissolved gas from the liquid or from the vaporization of the liquid by the laser heating. Then the microbubble kept on growing if the pump laser power was sustained. The temporal evolution of the microbubble can be monitored by the chargedcoupled device (CCD) and an example is given in Fig. 23. There was a sharp nonlinear increase in diameter at the beginning because a fast temperature rise was obtained when the laser was switched on. Then the diameter increased with time with a good linearity (R2  0.99) between 50s and 150s. When the diameter was large, the temperature rise became slow and gradually got balanced. Since the FSR

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Fig. 23 The temporal change of the microbubble diameter

changed linearly over certain range of heating time, detecting FSR in this time range would not influence on the calibration curve for sensing. The FSR can be expressed by FSR D 1 2 / (2nLeff ), with n and Leff as the refractive index and the effective length of the microbubble interferometer, respectively. The FSR is inversely proportional to the diameter of the microbubble so that it can be used to detect the generation of the microbubble. The growth rate of the microbubble would change when the ambient parameters change. The FSR of the microbubble at a fixed heating time was used as the sensing signal. As the diameter of the BoT interferometer increases with time, the free spectral range (FSR) decreases. By measuring the FSR, the temperature and flow rate sensing are demonstrated. For temperature sensing, the gas for microbubble mainly came from the vaporization of the liquid, similar to boiling the water by heating. For flow rate sensing, the gas mainly came from dissolved gas delivered together with the microfluid and can be refreshed as the fluid flows. High sensitivity was achieved thanks to the photothermal effect of CNT and the small volume of the microbubble. The reflective spectra from the BoT structure at heating time of 70s, 100s, and 140s are shown in Fig. 24a, while the ambient temperature was kept constant. The laser power for heating was 221 mW. The fringe contrast was not as high as solid fiber micro-interferometers but was sufficient to determine the free spectral range (FSR) of the interference spectra. A compromise between increasing the absorption by CNTs and keeping good interference fringes should be considered during the CNTs deposition. The FSR decreases with a good linearity (R2  0.98) as the microbubble grows in time (Fig. 24b). This work might open a door to the development of novel reconfigurable fiber optofluidic sensors. The temperature and flow rate were measured by detecting the FSR. The   deviation of FSR was calculated by D D F SR  F SR =F SR  100%, where F SR was the statistically averaged FSR over ten tests.

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Fig. 24 (a) The interference spectra of the fiber optofluidic microbubble-on-tip structure and (b) FSR versus heating time

Temperature sensing was demonstrated and the results are shown in Fig. 25. At each temperature, the 980 nm laser was turned on with a power of 146 mW, and the reflective spectra of microbubble was recorded after heating for 60s. Figure 25a showed the FSR changes as a function of temperature. Temperature was measured between 25 ı C and 45 ı C with good linearity (R2 ) of about 0.998 and sensitivity of 1146 pm/ı C. This range covers the temperature range of the human body and is significant for the organ-on-a-chip applications. By using the same probe, good repeatability was confirmed by ten times of tests at 25 ı C (Fig. 25b). For the microfluidic flow rate sensing, a range of 0–150 nL/min was achieved with a laser power of 221 mW (Fig. 26a). A good linearity (R2  0.997) is achieved, and the sensitivity is – 31 pm/(nL/min), which is determined by the slope of linear fitting. Ten times of tests were performed with a flow rate of 120 nL/min, and good repeatability was achieved, with a statistical deviation of ˙0.94% (Fig. 26b). The microfluidic flow takes away part of the laser-generated heat; therefore, the microbubble was supposed to grow slower at higher flow rate. However, the opposite

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Fig. 25 (a) FSR changes versus temperature with a heating time of 60s at 146 mW and (b) the repeatability of the same probe for ten tests

Fig. 26 (a) FSR changes as a function of flow rate with a heating time of 120s at 221 mW and (b) the repeatability of the same probe for ten tests

was surprisingly observed, that is, the microbubble grows faster at higher flow rate, leading to a smaller FSR at a fixed heating time (Fig. 26a). The potential explanation is that the dissolved air in the liquid was delivered more efficiently at higher flow rate, leading to a faster growth of the microbubble. Therefore, the air solubility was initialized by ultrasonic bathing in both temperature and flow rate sensing.

Concentration Sensor In order to further enhance the lifetime of the BoT sensor, gold nanofilm was coated on the fiber tip instead of CNT film. Beside the lifetime and stability enhancement, the gold nanofilm also enhanced the efficiency of microbubble generation. A fiber-coupled laser was used for heating the fiber tip, and single gas microbubbles can be photothermally generated. An easy-to-fabricate and easy-touse BoT concentration sensor was developed based on the BoT structure. Two methods were proposed to monitor the growth process as a function of the concentration. One was by the spectral detection, as described in the last section. After heating for 10s, the laser was switched off and recorded the reflective spectra

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from the microbubble using an optical spectrum analyzer (OSA). The microbubble acted as an interferometer based on the interference between the reflective beams from the two surfaces of the microbubble. The free spectral range (FSR) was measured so that the effective cavity length, L, can be further calculated as a function of the concentration. The other was diameter detection by imaging. The microbubble diameter was monitored by imaging, and the diameter difference between heating time of 2s and 7s was calculated as a function of concentration. The concentration of the solution can influence the growth rate of the gas microbubble in two different ways. One is the evaporation of liquid near the fiber tip, and the other is gas generated from the heat-induced chemical decomposition. Solutions of sucrose and hydrogen peroxide (H2 O2 ) were used as samples for testing, and the experimental results confirmed the high sensing performance, including a large dynamic range and high sensitivity. This study is the first to report the development of the lab on tip (LoT) based on BoT for concentration detection. This technique has many advantages such as high sensing performance, ease of fabrication and use, and low cost. The schematic structure of LoT technology is shown in Fig. 27a. The coated fiber tip was inserted into a glass capillary with a square cross section of 1 mm  1 mm, which was filled with deionized (DI) water or other kinds of liquid samples. Gas dissolved in the liquid may accelerate the generation speed of the microbubble. Therefore, the dissolved gas was removed by processing the liquid in an ultrasonic bath for 15 min. The liquid was withdrawn into the capillary by using a syringe pump or through the capillary force. During the experiment, the liquid was kept static. A 1550 nm laser was coupled into the fiber and heated the liquid near the core of the fiber tip. This wavelength was chosen because of its strong absorption in DI water. When the laser was switched on, a microbubble can be generated on the tip, and its diameter, d, increases with time, as shown in Fig. 27b. The growth process was monitored using an optical microscope with a low-cost CCD camera. The growth rate was determined to be about 11.61 m/s. At 10s, the laser was switched off, and the microbubble remained stable for a relatively long time.

Fig. 27 (a) The schematic structure and (b) the temporal growth of LoT

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Fig. 28 Sucrose concentration detection based on (a) imaging and (b) spectral detection

Fig. 29 (a) Microscopic images of the microbubbles and (b) the diameter difference as a function of H2 O2 concentration

Figure 28 shows the performance of sucrose concentration sensing with a dynamic range of two orders of magnitude, from 0.5 wt% to 50.0 wt%. The diameter

d and the FSR L are shown as a function of the sucrose concentration, which was obtained by imaging and the spectral method, respectively. Results are shown in the log-log scale for clarifying the details in the lower concentration range. The diameter of the microbubbles was monitored after a fixed heating time. Obviously, d decreases with the increase of sucrose concentration. The results can be explained as below. As sucrose concentration increases, its boiling point increases, and it takes longer time to accumulate heat for the liquid near the tip to vaporize. Linear fits are also given, both with a linearity of 0.96. The maximum sensitivity was 5.1 m/% at a concentration of 0.5 wt%. The averaged standard deviations were 0.6 m (5.6%) and 1.3 m (2.8%), respectively. As H2 O2 concentration increases, there are two contrary factors that affect the growth of the microbubble: (1) Similar to that for the sucrose solution, it takes a longer time for the liquid near the tip to vaporize at a higher concentration. Thus, the growth speed of the microbubble tends to decrease. (2) When heated by a laser, H2 O2 decomposes and generates oxygen (O2 ) gas. More gas can be generated

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with a higher concentration of H2 O2 , which tends to increase the growth speed of the microbubble. The results confirmed that the latter factor dominated in the experiment, which in principle was different from sucrose sensing. Therefore, the diameter difference increases as a function of H2 O2 concentration. For H2 O2 detection, the imaging method was chosen due to low cost. The calibration results are shown in Fig. 29. The microscopic images of the microbubbles generated with different concentrations of H2 O2 were recorded at different heating times (Fig. 29a). d as a function of the H2 O2 concentration is shown in the loglog scale in Fig. 29b. A maximum sensitivity of 93.84 m/M was obtained. The concentration sensing based on BoT structure achieved an unprecedented dynamic range of five orders of magnitude, from 105 M to 1 M.

Conclusion In this chapter, optical fiber microfluidic sensors based on two types of optophysical effects, i.e., laser-induced force and photothermal effect, are introduced. The laser-induced force is introduced for the application in optical tweezers and optical manipulation of microparticles. Controllable optical manipulation based on both graded-index fiber and single-mode fiber, either with a taper or flat endface, can be achieved by introducing a tunable microcavity or by adjusting the strain on GIF. Flow rate sensing based on the optical fiber manipulation methods can achieve a large dynamic range and high sensitivity. Photothermal effects, associated with the laser-induced heat and its transfer in a microring resonator, are introduced for microfluidic flow rate sensors. In addition the technique that gas microbubbles are generated on a fiber tip and used as a micro-interferometer is introduced for flow rate and temperature sensing, as well as concentration sensing with a large dynamic range. The optical fiber microfluidic sensors have distinct advantages including easy to fabricate, miniaturization, low cost, high sensitivity, and large dynamic range. Besides the parameters such as flow rate, temperature, and concentration, as introduced in this chapter, we believe that many new optical fiber microfluidic sensors will be developed for other physical, chemical and biological parameters in the future.

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamentals of MNFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabrication of MNFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optical Properties and Opportunities of MNFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamentals of Microfluidics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabrication of Microfluidic Chips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manipulation of Fluids in Microchannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Planar Microfluidic Chip-Based Biconical MNF Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . Refractive Index Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evanescent-Wave Absorption Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evanescent-Wave Fluorescence Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nanoparticle Sensors Based on the Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low Refractive Index Polymer-Coated Coiled MNF Sensors . . . . . . . . . . . . . . . . . . . . . . . . Evanescent-Wave Absorption Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Refractive Index Sensor Based on Coiled MNF Resonator . . . . . . . . . . . . . . . . . . . . . . . . Refractive Index Sensor Based on Coiled MNF Grating . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Rate Sensor Based on Coiled MNF Coupler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capillary-Based MNF Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Refractive Index Sensor Based on Liquid Core Optical Ring Resonator . . . . . . . . . . . . . . Biomolecular Detection Based on Liquid Core Optical Interferometer . . . . . . . . . . . . . . . Gold Nanoparticles Functionalized MNF Localized Surface Plasmon Resonance Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Refractive Index and Label-Free Biochemical Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cancer Biomarkers Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Optical micro-/nanofiber (MNF) microfluidic sensors – the synergistic integration of MNF and microfluidics – provide a number of unique characteristics for enhancing the sensing performance and simplifying the design of microsystems. With diameter close to or below the wavelength of guided light and high index contrast between the MNF and the surrounding, an MNF shows a variety of interesting waveguiding properties, including widely tailorable optical confinement, evanescent fields, and waveguide dispersion. MNF sensor has been attracting increasing research interest due to its possibilities of realizing miniaturized fiberoptic sensors with small footprint, high sensitivity, fast response, high flexibility, and low optical power consumption. Note that most of the MNF sensors used MNFs suspended in air or mounted in a bulky volume flow chamber; thus, surface contamination and environmental factors are likely to affect the stability of these sensors. Microfluidics is the science and technology of systems that process or manipulate small amounts of fluids, using microchannels with dimensions of tens to hundreds of micrometers. The microfluidic chip can provide natural protection of the MNFs, small volume of samples, and new sensing mechanisms. The fusion of MNFs and microfluidics opens a door to the practical application of MNF sensors. This chapter describes the fundamentals of MNFs and microfluidics and reviews recent progress in MNF microfluidic sensors regarding their fabrication, waveguide properties, sensing structures, and sensing applications.

Introduction In the past decades, optical fibers with diameters larger than the wavelength of transmitted light had quickly found extensive applications including chemical sensors, biosensors, and physical sensors (Lee 2003; Fan et al. 2008; Wolfbeis 2008). The optical fiber sensors have certain advantages that include immunity to electromagnetic interference, lightweight, small size, high sensitivity, large bandwidth, and ease in implementing multiplexed or distributed sensors (Lee 2003). Recently, with the rapid progresses in nanotechnology, there is an increasing demand for faster response, smaller footprint, higher sensitivity, and lower power consumption, which spurred great efforts for miniaturization of fiber-optic components and devices (Wu and Tong 2013). Although optical fibers with diameters close to the wavelength of propagating light had been fabricated, they had not been paid much attention until 2003. Tong and Mazur demonstrated low-loss optical waveguiding in micro-/nanofibers (MNFs) with diameters far below the wavelength of the guided light, which renewed research interests in MNFs (Tong et al. 2003). As potential building blocks for miniaturization of optical components and devices, MNFs have been attracting intensive research interests regarding their fabrication, properties, and applications. Among various MNF applications, optical sensing has been one of the most attractive research fields due to its possibilities of realizing miniaturized fiber-optic sensors with small footprint, high sensitivity, fast response,

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high flexibility, and low optical power consumption (Guo et al. 2014). Typical microfiber-based sensing structures, including biconical tapers, optical gratings, circular cavities, Mach–Zehnder interferometers, and functionally coated/doped microfibers, have been designed for refractive index, concentration, temperature, humidity, strain, and current measurement in gas or liquid environments (Lou et al. 2005). The manipulation of fluids in channels with dimensions of tens of micrometers – microfluidics – has emerged as a distinct new field (Whitesides 2006). Microfluidics has demonstrated the capability to influence subject areas from analytical chemistry to optics and information technology (Psaltis et al. 2006; Fan and White 2011). Although quite a few commercialized biosensors based on microfluidics have been developed for clinical diagnosis and biological analysis, microfluidics is still at its early stage of development. For example, absorbance detection is the most universal detection mechanism in analytical chemistry. However, this popularity has not translated to the microfluidics mainly because of the poor sensitivity due to the shallow channel depth and the difficulties in coupling the light into and out of these channels. The synergistic integration of MNF and microfluidics is a new sensing platform that provides a number of unique characteristics for enhancing the sensing performance and simplifying the design of microsystems. This chapter reviews recent progress in MNF microfluidic sensors regarding their fabrication, waveguide properties, and sensing applications. Finally, this chapter is concluded with an outlook for challenges and opportunities of optical MNF microfluidic sensors.

Fundamentals of MNFs For optical waveguiding, excellent geometric uniformity and surface smoothness of the MNFs are critical for achieving low optical loss and high signal-to-noise ratio, and therefore the fabrication process of these tiny fibers is vitally important. Compared with many other techniques such as photo- or electron-beam lithography, chemical growth, and nano-imprint, high-temperature taper-drawing method yields MNFs with lowest surface roughness, largest length, and excellent diameter uniformity. Also, the amorphous structure of the glass material bestows the MNF with circular cross section, which is ideal for obtaining waveguiding modes by solving Maxwell’s equations analytically. This section briefly introduces the fabrication of MNFs and guiding properties of MNFs.

Fabrication of MNFs Flame-heated taper-drawing is widely used to draw MNFs from standard optical fibers. A typical illustration of flame-heated taper-drawing process is shown in Fig. 1a. A hydrogen–oxygen flame is used for heating the fiber. Under a certain pulling force, the fiber is stretched and elongated gradually with reduced diameter

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Fig. 1 (a) Schematic diagram of flame-heated taper-drawing of an MNF from a standard optical fiber (Adapted from Wu and Tong (2013), with permission from the Walter De Gruyter GmbH) (b) Transmission spectra (offset) of tapered microfibers with diameters of 8 m, 4 m, 2.5 m, and 1.2 m (corresponding to tapering lengths of 10 mm, 14 mm, 15 mm, and 16 mm) and (c) measured FSR monotonously decreases with the fiber diameter (Adapted from Li et al. (2014a), with permission from Elsevier)

until the desired length or diameter of the fiber taper is reached. Using this technique, the as-fabricated MNF is usually attached to the standard fiber through the tapering region at both ends and is usually named as a “biconical” fiber taper. Importantly, it is critical to in situ monitoring the waveguiding properties of the MNF during the pulling process in terms of propagation loss, multimode interference, and group velocity delay by measuring optical transmission via standard fiber. Typically, when the taper length is from 10 to 16 mm, the total transmittance of the multimode waveguiding light is higher than 80% around 1550-nm wavelength. By launching broadband light from a white light source (halogen lamp) into one side of the as-fabricated MNFs and collecting the output from the other side with an optical spectrum analyzer, the broadband transmission spectra of the microfibers with different diameters are measured within the wavelength range of 1500–1600 nm (Fig. 1b). For those microfibers with diameters larger than 1.2 m (the singlemode cutoff diameter for an air-cladding silica microfiber operating at 1.55 m wavelength), higher-order modes are supported. Experimentally, for microfibers with diameters larger than 2 m, evident sinusoidal oscillatory transmission features are clearly seen. While for those with diameters close to or thinner than 1.2 m, only the fundamental HE11 mode is supported, and thus the sinusoidal oscillation disappears. The relationship between the free spectrum range (FSR) and the diameter of the microfiber is shown in Fig. 1c. It shows that the FSR monotonously decreases with the fiber diameter (Li et al. 2014a).

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Fig. 2 Typical images of glass MNFs. (a) A 1.5-m-diameter silica glass microfiber with a bending radius of 100 m. (b) A 1-mdiameter silica glass microfiber decorated with gold nanorods. (c) A coiled 1-m-diameter silica glass microfiber with a total length of about 1 cm. (d) An 800-nm-diameter nanofiber integrated with a microfluidic chip with a 5-m-wide detection channel

In some situations, flame-heated systems may present disadvantages such as the random turbulence of the flame and oxygen requirement in the burning process, leading to H2 O/OH contamination in MNFs. To avoid these issues, a CO2 laser beam can be used as an alternative heating source. Usually, the direct laser heating tapering procedure shows a self-regulating effect, which automatically stops the stretching process when the fiber diameter goes down to a certain value (usually above 1 m). By drawing MNFs in a microfurnace comprising a sapphire tube heated with a CO2 laser, Sumetsky successfully fabricated sub-m-diameter MNF with excellent surface smoothness and diameter uniformity (Sumetsky et al. 2004). Besides the abovementioned techniques, electrically heated taper-drawing approach is another simple and effective technique for fabricating high-quality MNFs (Shi et al. 2006). Usually, the electrical heater can be shaped into various geometries to precisely generate required temperature and temperature distribution, which makes it possible to draw MNFs with more flexibilities. In order to bestow the MNFs with greater versatilities, a number of postfabrication techniques including micromanipulation, plastic bend, coating, and embedding have been investigated in the past years. For reference, Fig. 2 shows typical images of twisted MNF, coiled MNF, gold nanorod-decorated MNF, and microfluidic chip embedded MNF, respectively.

Optical Properties and Opportunities of MNFs Comparedwith a conventional optical fiber, a high index-contrast (n) MNF with wavelength (œ) or subwavelength (sub-œ) diameter offers a number of interesting

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optical properties and opportunities, including tight optical confinement, strong evanescent field, and small mass/weight (Guo et al. 2014). For basic investigation, a straight MNF is assumed to have a circular cross section, a smooth sidewall, a uniform diameter, and an infinite cladding with a step-index profile. The fiber diameter (D) is not very small (e.g., D > 10 nm) so that the statistic parameters permittivity (") and permeability () can be used to describe the responses of a dielectric medium to an incident electromagnetic field. By solving Maxwell’s equations, one can obtain the field distribution of a waveguiding MNF. Figure 3 shows power distribution (Z-direction Poynting vectors) of HE11 mode of silica MNFs with diameters of 800, 400, and 200 nm in 3D and 2D view, respectively. It is clear that, while an 800nm-diameter MNF confines major energy inside the fiber, a 200-nm-diameter MNF leaves a large amount of light ( > 90%) guided outside as evanescent waves. Therefore, a freestanding MNF offers an opportunity to waveguide tightly confined optical fields with a high fraction of evanescent fields, which distinguishes the MNFs from many other waveguiding structures. For example, in Fig. 3b, the 400-nm-diameter nanowire guides a 633-nm-wavelength light with about 30% power outside as evanescent waves, while it maintains an effective mode area of about 560 nm in diameter.

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Tight optical confinement bestows the MNF with small allowable bending radius (e.g., low loss when passing through sharp bends) and small mode area, which makes MNFs highly potential for compact circuits and devices with smaller footprint, faster response, and lower power consumption. As shown in Fig. 4a, b, Li et al. demonstrated Mach–Zehnder interferometers (MZIs) assembled with tellurite glass MNFs, achieving good interference fringes with extinction ratios of 10 dB (Li and Tong 2008). Gu et al. reported highly versatile nanosensors using polymer single nanowires as shown in Fig. 4c. The reversible response of the polyacrylamide (PAM) nanowire is tested by alternately cycling 75% and 88% RH air inside the chamber, achieving an excellent reversibility as shown in Fig. 4d. The estimated response time (baseline to 90% signal saturation) of the sensor is about 30 ms, which is one or two orders of magnitude faster than those of existing RH sensors (Gu et al. 2008).

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Fig. 4 MZI assembled with tellurite glass MNFs and humidity sensor assembled with PAM nanowire and fiber tapers. (a) Optical microscope image of an MZI assembled with two 480-nmdiameter tellurite MNFs. (b) Transmission spectrum of the MZI shown in Fig. 4a (Adapted from Li and Tong (2008), with permission from Optical Society of America). (c) Schematic illustration of the polymer single-nanowire humidity sensor. Inset, optical microscope image of an MgF2 supported 410-nm-diameter PAM nanowire with a 532-nm-wavelength light launched from the left side. The white arrow indicates the direction of light propagation. (d) Reversible response of the nanowire tested by alternately cycling 75% and 88% RH air (Adapted with permission from Gu et al. (2008). Copyright (2008) American Chemical Society)

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Fig. 5 Schematic illustration of a waveguiding MNF for optical sensing. The sample light interaction (e.g., scattering, absorption, and emission) is reflected by the output (e.g., intensity, phase, or spectrum) of the MNF (Adapted with permission from Guo et al. (2014). Copyright (2014) American Chemical Society)

Strong evanescent field offers strong near-field interaction between the MNF and its surroundings, making the MNF highly favorable for optical sensing. The mechanism of a typical MNF sensor is schematically illustrated in Fig. 5. Within the optical near field of an optical MNF, a slight fluctuation in dielectric constant might evidently modify the output of the MNF by scattering, absorption, and emission of the guided light. By measurement of the intensity, phase, polarization, or spectrum of the output, the cause behind the fluctuation can be identified (Guo et al. 2014). In most situations, the measurand is either spatially distributed or highly localized. For sensing spatially distributed samples, which are usually in forms of or dissolved in liquids or gases, a typical approach is measuring the refractive index or absorption, which directly reflects the change of measurands (e.g., concentration, temperature, or pressure). The high fractional evanescent fields that are directly exposed to the surrounding sample may greatly enhance the sensitivity of the MNF sensors. Categorized by measurand, MNF sensors for refractive index, temperature, humidity, strain, and current measurement in gas or liquid environments have been reported (Lou et al. 2005).

Fundamentals of Microfluidics Microfluidics, the manipulation of liquids in channels with cross-sectional dimensions on the order of 10–100 m, has been a central technology in a number of miniaturized systems that are being developed for chemical, biological, medical, and optical applications (Arora et al. 2010). With the fusion of photonics and microfluidics, optofluidics has emerged as a distinct new field. Integration and reconfigurability are two major advantages associated with optofluidics. It provides new freedom to both photonics and microfluidics and permits the realization of optical and fluidic property manipulations at microscale. In the past decade, optofluidics has evolved into a multidisciplinary field, including chemical and biological sensing, lens-free imaging, particle manipulation, and tunable optical

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Fig. 6 Scheme describing replica molding of microfluidic chip. (a) A master with designed pattern is fabricated by lithography. (b) The prepolymer is cast on the master and cured. (c) The PDMS replica is removed from the master. (d) Exposing the replica and an appropriate material to an air plasma and placing the two surfaces in conformal contact make a tight, irreversible seal

devices, such as liquid core–liquid cladding waveguide, microlenses, gratings, and light source (Monat et al. 2007; Chen et al. 2012).

Fabrication of Microfluidic Chips To date, a number of materials (e.g., glass, silicon, poly(dimethyl siloxane) (PDMS), and poly(methyl methacrylate) (PMMA)) have been used for fabricate microfluidic chips. PDMS is one of the most widely used materials to fabricate microfluidics, lab-on-a-chip, microelectromechanical systems, and flexible electronics/photonics device due to its high transparency, low refractive index, high flexibility, and high Poisson’s ratio coefficient. Although any microfluidic fabrication method can, in principle, be adapted to fabricate a device for optical application, most of the implementations thus far have been with soft lithography. Soft lithography provides an approach to rapid prototyping of both microscale and nanoscale structures and devices on planar, curved, flexible, and soft substrates especially when low cost is required (Qin et al. 2010) Fig. 6 describes a typical procedure for fabricating PDMS microfluidic chip. Briefly, a master with designed pattern is fabricated by lithography. The prepolymer is then cast on the master and cured. When the PDMS replica is removed from the master, the replica and an appropriate substrate are exposed to an air plasma, and placing the two surfaces in conformal contact makes a tight, irreversible seal.

Manipulation of Fluids in Microchannels In order to manipulate fluids in microchannels, one has to understand how fluids behave at microscale. In the microchannels, surface forces dominate over body and

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Fig. 7 Typical approaches to manipulate fluids in microchannels. (a) Pneumatic actuated peristaltic micropump. (b) Schematic illustration of electrowetting. (c) Schematic illustration of electroosmotic flow micropumps. (d) Schematic illustration of manipulation of fluid in microchannel via a syringe pump

inertial forces. As the channel dimensions decrease, the surface forces decrease proportional to the square of the length scale. The body forces, however, decrease as the cube of the length scale. Thus, a tenfold decrease in channel dimensions leads to a tenfold increase in the surface-to-volume ratio of the channel, 100-fold decrease in surface forces, and 1000-fold decrease in volume forces. Therefore, the surface forces become tenfold more important relative to the body forces. To date, a number of approaches have been used to manipulate fluid in microchannels, including peristaltic pumps, electrowetting, electroosmotic flow micropumps, negative pressure generated by syringe pump, and so on. Peristaltic pumps are based upon using a series of actuation membranes to displace volume in the desired flow direction (Unger et al. 2000). By cycling membrane displacement, a peristaltic motion and volume displacement may be achieved. Peristaltic micropumps require at least three actuation membranes in series to obtain a nonreversible pump stroke. A typical pneumatic actuated peristaltic micropump is shown in Fig. 7a. The maximum pumping rate of 2.35 nL/s was attained at pump cycles of 75 Hz; above this rate, increasing numbers of pump cycles compete with incomplete valve opening and closing. The pumping rate was nearly constant until above 200 Hz and fell off slowly until 300 Hz. Electrowetting is based upon a change in the liquid–solid surface tension by charging the electrical double layer at an electrode surface. Figure 7b shows a schematic illustration of manipulation of a droplet in microchannel via electrowetting technology. By electrically changing the wettability of each of the electrode on the surface, a droplet on these electrodes can be shaped and driven along the active electrodes. Currently, electrowetting mechanism has been used extensively for droplet manipulation and tunable microliquid prism (Lee et al. 2002, 2013).

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Electroosmotic flow (EOF) pumping is another subset of electrokinetic phenomena related to the movement of electric charges in an applied electric field (Manz et al. 1994). Charge movement produces a shear on the surrounding fluid to produce flow. Electroosmotic pumps are used in small channels without a need for high pressures and may be combined with electrophoresis in bioanalytical separations. Note that electroosmotic flow is sensitive to the surface charge, Debye length, and applied electric field. As shown in Fig. 7c, in the presence of the electric field, positively charged ions will move toward the negatively charged cathode, and the negatively charge ions will move toward the positively charged anode. In the bulk region of the microchannel, charge neutrality is maintained, so there is no net charge movement. Although the abovementioned techniques have been widely used in chip-based chemical synthesis, electrophoresis, and manipulation of droplets, their application in microfluidic sensors is quite limited owing to the complicated fabrication process and control system. Figure 7d shows a schematic illustration of manipulation of fluid in microchannel via a syringe pump; the negative pressure sampling method has been proven to be an efficient and convenient approach to manipulate small volume of fluid in the microchannel (Zhang et al. 2006). Two obvious advantages are their freedom from electrokinetic bias effects and their ease for fabrication and manipulation.

Planar Microfluidic Chip-Based Biconical MNF Sensors One of the most straightforward approaches to assemble an MNF sensor is using a biconical MNF, which has been successfully applied in measuring a variety of samples. Due to its simple configuration, it can be integrated with planner microfluidic chip with ease. This kind of sensors usually measures the transmission intensity of the MNF and retrieves the sample information by concentrationdependent optical intensity.

Refractive Index Sensor Measuring the refractive index of a sample is one of the most popular approaches in bio-/chemical analysis and label-free sensing. In particular, it is extremely useful for detecting samples that do not absorb in the UV-Vis range or without fluorescence. Because the refractive index signal scales with the analyte concentration or surface density, rather than with the total number of molecules, refractive index detection is attractive for microfluidic sensors that have extremely small detection volumes. An MNF microfluidic sensor, which measures the refractive index of liquids propagating in a millimeter-scale channel, was demonstrated in Polynkin et al. (2005). The MNF was fabricated from a commercial single-mode fiber and embedded in PDMS. The sensing device, which is illustrated in Fig. 8a, was fabricated as follows. A rectangular cuvette serves as a mold for PDMS. A slit was machined through the bottom of the cuvette to accommodate a 3-mm-thick glass rod that

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Fig. 8 (a) Illustration of the MNF sensor with liquid channel. (b) Measured optical transmission versus refractive index of the liquid in the channel for two sensors with different taper thicknesses at the waist: curve1–1.6 m; curve 2–700 nm. Circles indicate the measurement of data points; the arrow shows the refractive index of the surrounding polymer (Adapted from Polynkin et al. (2005), with permission from the Optical Society of America)

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defines the channel for the sample liquid in the sensor. When the rod was in place, it was sticking out of the bottom of the cuvette by approximately one half of its diameter. Thus the resulting 1-cm-long and 3-mm-wide channel has a semicircular cross section. After the mold was filled with liquid PDMS, the MNF oriented orthogonally to the glass rod was slowly lowered into the uncured PDMS from the top. After the PDMS is cured at room temperature, the device was flipped over and the glass rod was removed, exposing the channel for the sample liquid. The sensing mechanism of this sensor is based on the fact that the shape of the fundamental optical mode, which travels through the sensing channel, was modified by the refractive index contrast between the sample liquid and the PDMS channel wall. When the refractive index of the sample liquid is higher than that of the PDMS (1.402 at 1.5-m wavelength), the mode is attracted into the channel, where the light is effectively scattered because of the presence of the channel boundaries and is absorbed in the liquid. If the refractive index of the sample liquid is lower than the index of PDMS, the mode propagates mostly in the uniform, transparent polymer. Consequently, the change of the fundamental mode resulted in variation of the MNF transmission. The sensitivity of the sensor was investigated by measuring the optical transmission through the MNF with the channel filled with solutions of glycerol in water. The refractive index at 1.5-m wavelength of the solutions can be varied from 1.311 for pure water to 1.459 for pure glycerol by using different concentrations of glycerol. The resulting calibration curves are shown in Fig. 8b for two different sensors, fabricated with (1.6 ˙ 0.2) m-diameter and (700 ˙ 200) nmdiameter MNF. In both cases, the transmission is at maximum when the refractive

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index of the sample liquid matches that of the PDMS, and the thinner MNFbased sensing device is more sensitive, achieving a potential measurement accuracy of 5  104 .

Evanescent-Wave Absorption Sensor Absorbance detection is one of the most universal detection mechanisms in analytical chemistry. However, this popularity has not been translated to the microfluidic sensors as the on-chip measurement of absorbing species has proven to be challenging, mainly because of the shallow channel depth (short optical path length) in microfluidic chip and the difficulties in coupling the light into and guiding the light out of these channels. When light is guided along an MNF, the evanescent fields outside the MNF interact with the analytes nearby. The theory of evanescent wave absorption (A) is given by Lambert–Beer’s law: A D ’L˜ where ’ is the absorbance coefficient, L is the detection length, and ˜ is the fractional power of evanescent fields outside the MNF. Most previously reported evanescent field absorption sensors adopted tapered fibers with a typical waist diameter larger than 10 m, which results in low ˜ and subsequently low sensitivity. Owing to the large achievable L and ˜, MNF-based evanescent field absorption has high potential for high sensitivity. Note that a great number of the reported evanescent field absorption sensors used MNFs suspended in air or mounted in a bulky volume flow chamber; thus, surface contamination and environmental factors are likely to affect the stability of these sensors. Zhang et al. (2011) developed an MNF absorption sensor by using a 900nm-diameter silica nanofiber embedded in a 125-m-wide microchannel with a detection length of 2.5 cm. When an MNF was integrated with a microfluidic chip as shown in Fig. 9, the microfluidic chip can provide natural protection of the MNF and decrease the sample volume dramatically to 500 nL. Moreover, the surface of the MNF can be cleaned and renewed after a measurement by flushing water, ethanol, and air via the microchannels. The microfluidic chip was fabricated by bonding a PDMS replica to a piece of PDMS membrane to seal the microchannels. In order to embed an MNF into a microchannel, the width and depth of the microchannels were designed and fabricated to be 125 m and 150 m, respectively (Fig. 9b). The embedding process was manipulated by a 3D travel translation stage under an optical microscope. Two 2-cm-long capillaries (150 m i.d., 375 m o.d.) were inserted into the capillary channel; one served as sampling probe and the other connected to a syringe for sample delivery. As shown in Fig. 9c, a 1.5-m-diameter MNF guiding a 473-nm-wavelength laser was embedded in a microchannel. When 0.01 mM fluorescein solution was injected into the microchannel, bright fluorescence excited by the 473-nm-wavelength laser can be seen as shown in Fig. 9d, indicating evanescent field outside the MNF.

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Fig. 9 Integrated MNF microfluidic optical sensors. (a) Optical microscope image of a tapered fiber with a waist of 900 nm. (b) Optical microscope image of the cross section of a PDMS microchannel. (c, d) Microfluidic structure before and after fluorescence excitation. (e, f) Transmission spectra of MB (e) and CB-BSA (f) obtained at different concentrations using a 900-nm-diameter nanofiber (Adapted from Zhang et al. (2011) with permission from Royal Society of Chemistry)

Investigated by measuring the absorbance of methylene blue (MB) around 630nm wavelength, the sensor shows a detection limit down to 50 pM with excellent reversibility in a concentration range of 0–5 nM (Fig. 9e). The sensor has also been applied to bovine serum albumin (BSA) measurement, achieving a detection limit of 10 fg/mL (Fig. 9f). Note that the probing light power was about 150 nW, suggesting a promising route to low-power high-sensitivity biochemical sensors.

Evanescent-Wave Fluorescence Sensor The evanescent wave fluorescence sensor is a well-developed tool for a wide range of applications, particularly for biological sensing (Leung et al. 2007). Fluorescence photons which can be efficiently coupled into guided modes of optical nanofibers offer a possibility for single-molecule or single-nanoparticle sensing (Yalla et al. 2012). Li et al. (2015) developed an MNF fluorescence sensor by using

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Fig. 10 (a) Schematic illustration of the experimental setup for the biconical MNF-based fluorescence sensing system. (b) Fluorescence spectra of R6G at different concentrations. (c) Fluorescence spectra of QD-labeled streptavidin at different QD concentrations (Adapted from Li et al. (2015) with permission from MDPI, Basel)

a biconical MNF with a diameter of 720 nm for both excitation and fluorescence collection. Figure 10a shows the schematic experimental setup for fluorescence measurement. Excitation laser (532-nm wavelength) is coupled to the MNF from the left side of the unstretched fiber. A fiber spectrometer is used to record the fluorescence spectra. To enhance the signal-to-noise ratio, a 550-nm-wavelength short pass filter and a long pass filter (561 nm for rhodamine 6G (R6G) and 700 nm for quantum dot (QD)-labeled streptavidin) are used to remove any scattered or excitation light from the system. In the taper waist section (MNF), the mode of the unstretched fiber is adiabatically transformed into the strongly guided mode of the ultrathin section and back, resulting in a highly efficient coupling of light into and out of the taper waist. It is worth mentioning that a slot sample vial array was adopted for sample introduction because of its high sampling throughput (Du et al. 2005). The sample vials were produced from 0.5 mL centrifuge tubes by cutting a 1.5-mm-wide, 2-mm-deep slot at the conical bottom of the tube for pass-through of the sampling capillary. The slotted sample vials were horizontally fixed on a glass slide in an

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array, with the slot of each vial positioned horizontally to allow free passage of the sampling capillary through all the vial slots sequentially by linearly moving the glass slide along the direction of the array. The vials were filled alternately with 100 L samples and ultrapure water. Thus, the microchannel and the surface of the MNF can be cleaned before a new sample was introduced by linearly moving the glass slide. Samples in the slot vials were sucked into the microchannel by negative pressure generated by a syringe that connected with the sample outlet channel. In this case, the sampling throughput can be as high as 60 samples per hour. The performance of the sensor was systematically investigated by measuring the fluorescence intensity of R6G and QD-labeled streptavidin, respectively. When a 532-nm-wavelength laser was guided into a 720-nm-diameter MNF to excite the fluorescence, the fractional power outside the core is about 15%, resulting in a strong evanescent field for exciting the fluorescence and a high efficiency for collecting the fluorescence. In this case, the estimated fluorescence collection efficiency for the sensing system was about 13.5%. Note that decreasing the diameter of an MNF is an effective method to increase the fractional power outside the core; however, when the fractional power outside the core increases to a critical value, the fluorescence collection efficiency will decrease. As shown in Fig. 10b, the fluorescence intensity increases obviously with the increases of R6G concentration. The detection limit for R6G based on three times the standard deviation of the blank values was about 100 pM. Compared to conventional organic fluorophores, QDs are virtually immune to the effects of photo-bleaching, and have a relatively higher absorption cross section, which may further enhance the sensitivity. Figure 10c shows the typical fluorescence spectra of QD-labeled streptavidin at different QD concentrations ranging from 10 to 100 pM. The detection limit for QDs was about 10 pM. Because there are three to five streptavidins covalently attached on one QD, the detection limit for streptavidin was about 30–50 pM. With the advent of single-molecule detection and early disease diagnosis, there is a real need to reduce the detection volume with optical methods, because the concentrations of these samples are relatively high, but the amounts of these samples are quiet limit. To address this issue, total internal reflection with fluorescence correlation spectroscopy, confocal microscopy, zero-mode waveguides, and microstructured optical fibers have enabled the observation and/or detection of molecules and ions in reduced volumes ranging from atto- to nanoliter scale (Laurence and Weiss 2003). The abovementioned approaches have strengths in different areas and are suitable for different applications. However, they require expensive instruments and/or complicated fabrication processes, which prevent lots of researchers from studying and understanding biological processes on the molecular scale. MNF microfluidic sensors provide a cost-effective manner to achieve a small detection volume by manipulating the penetration depth of the evanescent field and the detection length of the MNF simultaneously (Zhang et al. 2015). To achieve a detection volume of 1.0 fl, a typical detection volume for singlemolecule analysis, it is necessary to reduce the channel width to less than 10 m. Meanwhile, to embed an MNF and deliver liquid samples, the other channels’

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Fig. 11 Schematic illustration of fabricating SU-8 master for replicating microfluidic chip with a sub-10-m-wide detection channel

dimension should be greater than that of the detection channel. It is a great challenge to fabricate a planar microfluidic chip with different channel depths, in particular, with a narrow and shallow detection channel by using standard lithography method. Figure 11 shows a four-step procedure for fabricating a SU-8 master for replicating PDMS microfluidic chip with a sub-10-m-wide detection channel. Firstly, a channel design without the detection channel on a mask is transferred onto a glass substrate (Fig. 11a). Secondly, a film of 10-m-thick SU-8 photoresist is spin-coated on the substrate, followed by soft bake, exposure, post exposure bake, and hard bake (Fig. 11b). Thirdly, a SU-8 microfiber is located onto the substrate by a 3D travel translation stage under an optical microscope. After post exposure bake and hard bake, the SU-8 microfiber attaches tightly to the SU-8 film (Fig. 11c). Finally, a film of 130-m-thick SU-8 photoresist is spin-coated on the substrate, followed by soft bake, expose from the bottom of the substrate, post exposure bake, developing, and hard bake (Fig. 11d). Figure 12a shows a typical optical micrograph of an as-fabricated MNF microfluidic chip with a 500-m-long, 8-m-wide detection channel, which connects two 150-m-wide channels for sample introduction. As shown in the inset of Fig. 12a, when 0.01 mM fluorescein solution was introduced into the 5-m-wide detection channel, and a 473-nm-wavelength laser was launched to the nanofiber, a bright fluorescence spot excited by the evanescent field outside the nanofiber,

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Fig. 12 (a) A typical optical micrograph of an MNF microfluidic chip with a nanofiber crossed a narrow detection channel. Inset: Optical micrograph of an 800-nm-diameter nanofiber crossed a 5-m-wide channel. A bright fluorescence spot excited by evanescent field indicates a detection length of 2.5 m. (b) Calculated fluorescence intensity versus the concentration of fluorescein. Insets: (1–5) Optical micrographs of the fluorescence spots excited by evanescent field outside an 800-nm-diameter nanofiber with fluorescein concentrations of 1  107 , 3  107 , 5  107 , 7  107 , and 1  106 M, respectively. Laser power, 5 mW; exposure time, 1.0 s. Scale bar: 5 m (Adapted from Zhang et al. (2015) with permission from Optical Society of America)

indicating an effective detection length of 2.5 m. When the penetration depth is defined as the length where the evanescent field intensity decays to 10% of the highest intensity outside the nanofiber, the penetration depth of an 800-nmdiameter silica nanofiber operated at 473-nm wavelength is about 150 nm, leading to an effective detection volume of  1.0 fl. To investigate the sensitivity and linearity of the sensor, fluorescein solutions with concentrations were measured by calculating the fluorescence intensity. A linear concentration-dependent response was obtained as shown in Fig. 12b. When an electron multiplying charge-coupled device (EMCCD) or a fast-response, high-resolution spectrometer is used to record the signal, the femtoliter-scale MNF microfluidic sensor can provide a compact and versatile sensing platform for sensitive and fast detection of ultra-low volume samples, as well as studying the dynamics of single nanoparticles or single molecule.

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Fig. 13 Basic model of the light scattering of a nanoparticle in the vicinity of a nanofiber. The minimum required number (Np) of particles for being detectable with respect to the particle diameter (Dp, min). The refractive index (n) of the particle and the diameter (D) of the fiber are assumed to be 1.5 and 200 nm, respectively (Adapted from Wang et al. (2007) with permission from Elsevier)

Nanoparticle Sensors Based on the Scattering Rapid detection and sizing of nanoparticles are becoming of increasing importance for applications in early-stage clinical diagnostics, process control of semiconductor manufacturing, and environmental monitoring. For MNF microfluidic sensor, the spectrum of scattered light provides an effective approach to detect nanoparticles with deep subwavelength cross section, which is imperceptible by ordinary light beams. It can reveal important information about the structure and dynamics of the material being examined. For example, the probing light guided along an MNF can be confined to a “thin light beam” with size comparable to the nanoparticles while maintaining strong evanescent fields for near-field interaction. In this case, a nanoparticle can cause detectable change to the transmission of an MNF by optimizing the wavelength of the probing light and the diameter of the MNF. Based on Rayleigh–Gans scattering theory, calculation showed that with a 325-nmwavelength probing light guided in a 200-nm-diameter silica nanofiber, a single 90-nm-diameter particle (index of 1.5) can be detected (Fig. 13), indicating the possibility for single-molecule detection (Wang et al. 2007). Experimentally, Yu et al. (2014) proposed a fast, compact, and inexpensive method to perform single-nanoparticle detection and sizing in an aqueous environment by using nanofiber pairs. By monitoring the step changes in the transmitted power of the fiber, detection and sizing of single polystyrene (PS) nanoparticles

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Fig. 14 (a) Schematic illustration of the nanofiber sensing system, where a pair of nanofibers is used. (b) A typical transmitted power of the fiber during a time interval of 10 s when the PS nanoparticles are binding to the surface of the nanofiber one by one. (c) The optical microscopy image of the nanofiber with single PS nanoparticles bound to its surface in an aqueous environment. (d) SEM image of a portion of the nanofiber, where PS nanoparticles (R D 100 nm) are bound on its surface (Adapted from Yu et al. (2014) with permission from Willey-VCH)

down to radius of 100 nm in an aqueous environment were demonstrated. Figure 14a shows a schematic illustration of the nanofiber sensing system. Since both nanofibers can be used to detect the nanoparticles, the sensing speed is doubled compared with a single nanofiber case. A pair of nanofibers with typical diameter down to 500 nm is fabricated simultaneously by heating and pulling a standard single-mode fiber; the two tapered parts possessing uniform diameter are then immersed in a sample channel filled with deionized water. A typical fiber transmission is shown in Fig. 13b, from which several obvious abrupt decreases in transmitted power can be seen. These step changes correspond to single PS nanoparticle binding events, also confirmed by the CCD image (Fig. 13c), in which each scattering point corresponds to one single nanoparticle bound to the surface of the nanofiber. The one-by-one attachment of the nanoparticles on the surface of the nanofiber was confirmed by the scanning electron microscope (SEM) image (Fig. 13d). In addition, sizing of nanoparticles with a single radius (100 nm) and of mixed nanoparticles with different radii (100 nm and 170 nm) has been realized, with the result agreeing well with theoretical predictions by Raleigh–Gans scattering model for certain-sized nanoparticles. Moreover, with plasmonic enhancement, the nanofiber holds the potential for the detection of single gold nanorods with much smaller sizes.

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Low Refractive Index Polymer-Coated Coiled MNF Sensors Compared to other optical waveguide sensors, a prominent advantage of MNF sensor is its larger area of interaction with the ambient medium and seamless connection to the light source and detector. Nevertheless, the fragileness of a freestanding MNF and surface containments restrict its applications. The design of low refractive index polymer-coated coiled MNFs can effectively increase the robustness of the device and provide more sensing mechanisms.

Evanescent-Wave Absorption Sensor Lorenzi et al. (2011) developed an in-line absorption sensor, where the analyte flows in a fluidic channel whose walls were made of absorption-responsive coiled microfibers. Figure 15a describes the fabrication procedure: (1) coiling of a biconical MNF onto an expendable PMMA rod, which connects two Teflon tubes,

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(2) embedding the PMMA rod and the coiled MNF with a Teflon solution and curing at 80 ı C for 20 min, and (3) dissolving the PMMA rod by dipping the device in acetone for 1 day. In order to improve mechanical stability, the device is anchored on a microscope glass slide and covered with an UV-curable acrylate polymer as low-loss protective glue. Figure 15b shows the optical loss as a function of bright blue concentration. Dashed blue line represents the expected optical loss. Note that the linear relationship is a good approximation only for low concentrations. At low concentrations, the optical absorption will be proportional to the number of molecules effectively dispersed in the solution; since the number of molecules attached onto the fiber surface is too small to give any contribution. On the contrary, as the molarity increases, more and more molecules attach onto the fiber surface and the resulting optical absorption is dominated by these molecules. The proposed device may find applications as absorption-responsive tube when small volumes are needed.

Refractive Index Sensor Based on Coiled MNF Resonator Resonating structures can provide a simple, cost-effective, and sensitive technology for refractive index measurement and label-free biosensing. MNFs are ideal elements for assembling high Q resonators because of their low loss and very large evanescent fields. Similar to the in-line absorption sensor mentioned above, the coiled MNF refractive index sensor was obtained by embedding MNF microcoil resonator with Teflon, followed by removing its supporting rod. Figure 16a shows a schematic of the cross section of the coiled MNF resonator. Because of the interface with the analyte, the mode properties are particularly affected by the microfiber radius (r) and the coating thickness (d) between the MNF and the fluidic channel. Based on calculations, the fundamental mode is the one with the largest propagation constant and the only mode that is well confined in the vicinity of the MNF. Generally, thinner MNFs and smaller d leads to a larger fraction of the mode propagating in the analyte. By measuring the wavelength shift, the sensitivity was about 40 nm/RIU.

Refractive Index Sensor Based on Coiled MNF Grating Optical fiber gratings are widely used in fiber-optic sensors. The standard UV writing technique is inconvenient for an MNF due to the disappearance of the germanium core and the decrease of the field overlap with the core. To data, focused ion beam (FIB) milling and femtosecond infrared irradiation have been reported for the fabrication of MNF gratings; however, the expensive equipment and complicated procedure restricted their applications. Xu et al. (2009) suggested a novel fluidic MNF grating by wrapping an MNF on a microstructured rod, followed by coating them with a low refractive index polymer. Combining current enabling technologies on microstructured optical fibers and MNFs, it is possible to obtain periodic refractive index distribution, leading to the coupling between forward and

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Fig. 16 (a) Schematic of the cross section of an OMCRS (Adapted from Xu and Brambilla (2008) with permission from AIP Publishing). (b) The equivalent planar structure; n1 , n2 , n3 , and n4 are the refractive indexes of the hole, the rod, the OFM, and the coating, respectively; ƒ is the distance between two adjacent holes; d1 and d2 are the thickness of the outer and inner walls of the one-ring microstructured rod; r is the hole radius (Adapted from Xu et al. (2009) with permission from Optical Society of America)

backward propagating waves. When a coordinate increasing along the microfiber is used, the surface corrugations experienced by the mode propagating along the curvilinear coordinate are similar to those experienced by a mode propagating straight in proximity of a conventional planar grating. Unfolding the coiled MNF and the microstructured rod, the equivalent MNF grating structure can be taken as a coated MNF on a planar substrate with air-hole corrugations (see Fig.16b). A refractometric sensor can be obtained by exploiting the holes of the microstructured rod as microfluidic channels. The sensitivity is highly dependent on thickness of the outer and inner walls and the diameter of the MNF. Generally, sensitivity increases with the decrease of the effective wall thickness or the diameter of the MNF, because the fraction of the mode field inside the fluidic channel increases. The calculated sensitivity can be as high as 1200 nm/RIU for a 400-nm-diameter nanofiber.

Flow Rate Sensor Based on Coiled MNF Coupler Using an MNF coupler, Yan et al. (2016) demonstrated a “hot-wire” microfluidic flow rate sensor by wrapping an MNF coupler around a gold-coated capillary. The MNF coupler was fabricated by fusing and tapering two commercial single-mode

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Fig. 17 (a) Schematic illustration of an MNF coupler. (b) Schematic illustration of an MNF coupler wrapping around the gold-coated glass capillary and embedding in the UV-curable adhesive. (c) Output spectra of the packaged MNF coupler sensor. (d) The wavelength shift with different microfluidic flow rates under different incident power values of 100, 150, and 197 mW (Adapted from Yan et al. (2016) with permission from Optical Society of America)

fibers (SMF-28, Corning, USA) with flame-brushing method. As shown in Fig. 17a, the MNF coupler includes one uniform waist region, two transition regions, and four pigtails. To fabricate a flow rate sensor as shown in Fig. 17b, a glass capillary (1 mm i.d.) with a wall thickness of 120 m was employed as a microfluidic channel. With vacuum-coating technology, a 100-nm-thick gold film was coated around the capillary. After that, an MNF coupler was wrapped around the capillary with one turn by a rotating stage. Finally, the sensing sections of the MNF coupler and the capillary were packaged with a low refractive index adhesive (EFiRON UVF PC-375, Luvantix). Figure 17c shows a typical output spectrum with multiple interference peaks of a packaged sensor. Each dip represents a characteristic wavelength of the MNF coupler, at which the power couples to another fiber at maximum efficiency. Due to the absorption of the evanescent field by gold film, heat is generated and temperature increases. When the microfluid flows through the channel, it takes away part of the heat and cools the MNF coupler, resulting in wavelength shift of the resonant peak. Generally, the larger the flow rate is, the more heat microfluid takes away, and the more the wavelength shifts. Based on calculation, when the flow rate is 3 L/s, almost all the heat is taken away, and the wavelength shifts reach a saturation value. Figure 17d shows a typical output spectrum of the wavelength shift with different microfluidic flow rates under

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different incident power values of 100, 150, and 197 mW. Due to the long-distance interaction and high-temperature sensitivity, the proposed microfluidic flow rate sensor shows an ultrahigh flow rate sensitivity of 2.183 nm/(L/s) at a flow rate of 1 L/s.

Capillary-Based MNF Sensors The capillary-based MNF sensors achieve dual use of the capillary as a sensor head and as a fluidic channel. As a result, it takes advantage of high sensitivity of ring resonators or interferometers and low sample consumption of microfluidic chips. In addition, they are highly compatible with well-developed capillary technologies for automated fluid delivery.

Refractive Index Sensor Based on Liquid Core Optical Ring Resonator Liquid core optical ring resonator (LCORR) utilizes a fused silica capillary to carry the aqueous sample and to act as the ring resonator (White et al. 2006). The wall thickness of the LCORR is controlled to a few micrometers to expose the whispering gallery mode (WGM) to the aqueous core. Figure 18a shows a conceptual illustration of a LCORR sensor array. The WGM of each constituent ring resonator is launched through horizontally arranged MNFs, while the aqueous samples are conducted by the vertically positioned capillaries (see Fig. 18b). The LCORR uses the evanescent field of the WGM in the core to detect the refractive index change near the interior surface. Moreover, with the transverse arrangement, it is relatively easy for LCORRs to be integrated into a 2D array for simultaneous analysis of multiple samples. To fabricate a LCORR, a fused silica capillary (r1 D 0.45 mm, r2 D 0.6 mm) was stretched under a flame until the outer radius reaches 35–50 m, followed by further etching the capillary with low concentrations of HF to the desired wall thickness. And then, the LCORR was positioned in contact with an MNF of approximately 3–4 m in diameter, as illustrated in Fig. 18b. Light from a tunable laser diode (980 nm) is coupled into the WGM through the evanescent coupling at the LCORR exterior surface. The tunable laser repeatedly scans across a wavelength range of approximately 100 pm. The WGM spectral positions are recorded at the output end of the taper. During the experiment, the liquid sample is delivered by a peristaltic pump and tubings attached to the LCORR. The Q factor for the 3 m sensor is 4.1  105 , which implies a WGM linewidth of 2.4 pm. Assuming one fiftieth of the linewidth can be resolved, the LCORR can theoretically detect a refractive index change of 1.8  105 RIU. Besides its bulk refractive index sensing, LCORR is capable of detecting molecules on the interior capillary surface (White et al. 2007). When BSA prepared in the PBS buffer was pumped through the LCORR, the WGM spectral

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Fig. 18 (a) Conceptual illustration of a LCORR sensor array. (b) Cross section of a LCORR sensing element. The inner radius and the wall thickness are r1 and r2  r1 , respectively (Adapted from White et al. (2006) with permission from Optical Society of America)

position shifts quickly to a longer wavelength and then levels off, indicating that the equilibrium is reached between the BSA molecules in solution and on the LCORR surface. With the increased BSA concentration, the equilibrium WGM shift increases and then becomes saturated when BSA concentration is higher than 200 nM. The experimental results were in good agreement with the theoretical prediction. Theoretically, the LCORR was capable of detecting BSA below 10 pM with sub-pg/mm2 mass detection limit. The development of surface-enhanced Raman scattering (SERS) detection has made Raman spectroscopy relevant for highly sensitive biological and chemical sensors. Despite the tremendous benefit in specificity that a Raman-based sensor can deliver, development of a lab-on-a-chip SERS tool has been limited thus far. Zhu et al. (2007) developed a SERS-based detection tool by utilizing an optofluidic ring resonator (OFRR) platform. The LCORR serves both as the microfluidic sample delivery and as a ring resonator, exciting the silver nanoclusters and target analytes as they pass through the channel. Using this OFRR approach, a measured detection limit is about 400 pM for R6G. The measured Raman signal in this case is likely generated by only a few hundred R6G molecules, which foreshadows the development of a SERS-based lab-on-a-chip bio-/chemical sensor capable of detecting a low number of target analyte molecules.

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Fig. 19 (a–c) Schematic illustration of the fabrication procedure of the capillary-based optofluidic sensor. (d) Scanning electron microscopic (SEM) image of the capillary-based optofluidic sensor in cross-sectional view. (e) Sensor gram for surface activation and miRNA-let7a detection with a bulk concentration of 20 M: (1) water-DEPC rinsing for 30 min, (2) APTES silanization, (3) glutaraldehyde cross-linking, (4) immobilization of amine-modified DNA probes, (5) hybridization with miRNA-let7a. Inset: Measured transmission spectra before and after miRNA hybridization. (f) The kinetic binding curves of the miRNA-let7a with different concentrations (Adapted from Liang et al. (2017) with permission from Elsevier)

Biomolecular Detection Based on Liquid Core Optical Interferometer See Fig. 19. Optical interferometer has attracted increasing research interest in biosensing owing to its high sensitivity, simple structure, and label-free sensing capability. By measuring the resonance peak shift, a number of biomaterials has been detected by various optical interferometers. However, most of the optical interferometers are limited to laboratory conditions due to the complexity in experiment setup or the difficulties in sample pretreatment. To design a robust and portable optical

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interferometer sensor for clinical applications, Liang et al. (2017) developed a liquid core optical interferometer by tapering silica capillary and optical fiber in parallel. Figure 19a–c schematically shows the fabrication procedure of the sensor. Briefly, a bare single-mode fiber is aligned with a silica capillary in lateral contact in the same direction. Both the single-mode fiber and the capillary are heated by a flame and stretched by two fiber holders. The tapered structure is then embedded in a low refractive index polymer and packaged between two cover-slides for better spatial stability. Figure 19d shows a typical as-fabricated attached MNF and capillary. Owing to the extremely thin wall, light guided in the MNF can penetrate the capillary and interact with the fluid in the capillary, leading to a spectral shift with a response to biomolecule binding event. The resonance wavelength shows an obvious shift during the hybridization with miRNA-let7a (see Fig. 19e). The kinetic binding curves of the miRNA-let7a with different concentrations are shown in Fig. 19f. With the pre-immobilization of DNA probes, the biosensor is capable of detecting single-stranded microRNA-let7a (molecular weight: 6.5 k). A loglinear response from 2 nM to 20 M and a minimum detectable concentration of 212 pM (1.43 ng/mL) have been achieved. The sensor is promising for biomarker detection in preclinical applications owing to its advantages of high resistance to environmental perturbations, improved portability, easy fabrication and handling, and intrinsic connection to fiber-optic measurement.

Gold Nanoparticles Functionalized MNF Localized Surface Plasmon Resonance Sensors The exploitation of localized surface plasmon resonance (LSPR) of noble metal nanoparticles in the development of plasmonic sensors has attracted considerable research interest for many years. LSPR is the collective oscillation of conduction electrons confined to metal nanoparticles, whose resonance frequency has been shown to be strongly dependent on the particle’s size, shape, composition, and the dielectric properties of surrounding medium. Benefited from the favorable optical waveguiding properties of the MNFs, the LSPR of the noble metal nanoparticles could be effectively excited by a waveguiding approach. The highly efficient photonto-plasmon conversion of noble metal nanoparticles on the surface of an MNF may greatly facilitate and enhance light-matter interactions within a highly localized area and open up opportunities for developing sensitive sensors with miniaturized sizes, high compactness, and low optical power consumption.

Refractive Index and Label-Free Biochemical Sensor Lin et al. (2012) demonstrated an MNF LSPR sensor for refractive index sensing and label-free biochemical detection. The sensing strategy relies on the interrogation of the transmission intensity change due to the evanescent field absorption of

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immobilized gold nanoparticles (GNPs) on the MNF surface. Experimentally, the MNF was manufactured by tapering a standard single-mode fiber through a hydrogen–oxygen flame-brushing technique. The average waist diameter and waist length of the as-fabricated MNF was 48 m and 1.25 mm, respectively. To immobilize GNPs on the MNF surface, the MNF was cleaned by a threestep process of ultrasonic bath in acetone for 20 min, soaking in Piranha solution for 30 min, followed by a thorough flushing with ultrapure water, and drying in an oven at the temperature of 70 ı C. The clean MNF was then immersed in 1% solution of 3-mercaptopropyltrimethoxysilane (MPTMS) in ethanol overnight, leading to the formation of a thiol-terminated self-assembled monolayer (SAM) of MPTMS on the MNF surface. The thiol-functionalized MNF was subsequently rinsed by ethanol to remove unbound monomers from the surface and blowdried with nitrogen gas. Afterward, the MNF was submerged in the solution of GNPs to form a monolayer of GNPs on the MNF surface. Finally, GNPs functionalized MNF which was mounted in a flow cell for refractive index sensing and label-free biochemical detection. When sucrose solutions with various refractive indexes ranging from 1.333 to 1.403 were successively pumped into the flow cell, the transmission at the peak wavelength exhibited obvious decreases and the peak red-shifted as the surrounding refractive index alters from low to high. The refractive index resolution based on the interrogation of transmission intensity change is calculated to be 3.2  105 RIU. To further investigate the feasibility of the MNF LSPR sensor for label-free biochemical detection, N-(2,4-dinitrophenyl)-6-aminohexanoic acid (DNP) was chemically functionalized on the MNF surface as the molecular recognition probe, and anti-DNP antibody was employed as analyte, achieving a LOD of 1.06  109 g/ml for anti-DNP antibody.

Cancer Biomarkers Sensor Sensitive and selective detection for cancer biomarkers is critical in cancer clinical diagnostics. Li et al. (2014b) developed a novel MNF microfluidic biosensor using GNPs as amplification labels for the detection of alpha-fetoprotein (AFP) in serum samples. The main mechanism of this biosensor is the selective absorption of evanescent wave by GNPs when they approach the MNF surface. The sandwich assay shown in Fig. 20a was used for the detection of AFP molecules. First antibody was immobilized on the surface of the MNF as the capture antibody. Secondary antibody-functionalized GNPs were used as signal amplifiers. Owing to a prominent evanescent field with a depth of about several hundreds of nanometers around the MNF, GNPs binding to the MNF surface via specific interaction can lead to a great decrease in the output light intensity. Figure 20b, c shows a schematic illustration and photograph of an MNF sensor with an integrated PDMS chamber for sample delivery, respectively. The microfluidic chip can dramatically decrease sample consumption and improve the stability of the MNF sensor. To investigate the feasibility

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Fig. 20 (a) Schematic diagram of the immunoassay for ’-fetoprotein (AFP) detection using GNPs as signal amplification labels. (b) Schematic diagram of an MNF sensor with an integrated PDMS chamber for sample delivery. (c) Image of a sensor cell and SEM image of a 1.0-mmthick optical microfiber. (d) Secondary antibody-functionalized GNP enhanced sensor response to bovine serum samples spiked with different concentrations of AFP (Adapted from Li et al. (2014b) with permission from Elsevier)

of this sensing strategy for clinical applications, bovine serum samples spiked with different concentrations of AFP ranging from 0.2 to 1000 ng/mL was tested using the sandwich assay. Real-time response curves are shown in Fig. 20d. The limit of detection of this sensor for AFP is 0.2 ng/mL in PBS and 2 ng/mL in bovine serum, which is comparable to conventional assays. The advantages of this biosensor are simple detection scheme, fast response time, and ease of miniaturization, which might make this biosensor a promising platform for clinical cancer diagnosis and prognosis.

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Conclusion This chapter reviewed applications of MNF microfluidic sensors. It is shown that the MNF can serve as a basic sensing block for assembling compact and robust sensors. The fusion of microfluidic chip and MNFs provides a number of attractive advantages for enhancing the sensing performance and simplifying the design of microsystems. Compared with the sensors used, MNFs suspended in air or mounted in a bulky volume flow chamber, MNF microfluidic sensors are more stable, need less sample, and can be reused by cleaning the MNF surface via the microchannels. More importantly, the integration of MNF and microfluidics provides a new platform to develop novel sensing mechanisms and practical sensing structures for chemical, biological, medical, and optical applications. As a future outlook, there are a number of opportunities and challenges for MNF microfluidic sensing, including (1) higher sensitivity, detection of single molecule or single nanoparticle; (2) better selectivity, for example, enabling selective detection of target molecule by properly functionalizing the MNF structure with antibody; and (3) better robustness: as a tiny structure highly sensitive to environmental changes (e.g., temperature and displacement), long-term stability is highly desired for practical applications.

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All Optical Fiber Optofluidic or Ferrofluidic Microsensors Fabricated by Femtosecond Laser Micromachining

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Hai Xiao, Lei Yuan, Baokai Cheng, and Yang Song

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Femtosecond Laser and Material Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Free Electron Plasma Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy Deposition and Material Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propagation Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laser Processing System for Direct Writing in Optical Fiber . . . . . . . . . . . . . . . . . . . . . . . . Femtosecond Laser Micromachining System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Femtosecond Laser Writing in Optical Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid-Assisted Laser Processing in Optical Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to Liquid-Assisted Laser Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabrication of 3D Hollow Structure in Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . All-in-Fiber Optofluidic Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operation Principle and Sensing Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensor Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement of RI with the Optofluidic Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fiber In-Line Ferrofluidic Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Research and development in photonic micro-/nanostructures functioned as sensors have experienced significant growth in recent years, fueled by their broad applications in the fields of physical, chemical, and biological quantities. Compared with conventional sensors with bulky assemblies, recent progress

H. Xiao () · L. Yuan · B. Cheng · Y. Song Department of Electrical and Computer Engineering, Center for Optical Materials Science and Engineering Technologies (COMSET), Clemson University, Clemson, SC, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2019 G.-D. Peng (ed.), Handbook of Optical Fibers, https://doi.org/10.1007/978-981-10-7087-7_63

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in femtosecond (fs) laser three-dimensional (3D) micromachining technique has been proven an effective way for one-step fabrication of assembly-free microstructures in various transparent materials (i.e., fused silica). When used for fabrication, fs laser has many unique characteristics, such as negligible cracks, minimal heat-affected zone, low recast, high precision, and the capability of embedded 3D fabrication, compared with conventional long pulse lasers (i.e., ns laser). The merits of this advanced manufacturing technique enable the unique opportunity to fabricate integrated sensors with improved robustness, enriched functionality, enhanced intelligence, and unprecedented performance. Recently, fiber-optic sensors have been widely used in many application areas, such as aeronautics and astronautics, petrochemical industry, chemical detection, biomedical science, homeland security, etc. In addition to the well-known advantages of miniaturized in size, high sensitivity, immunity to electromagnetic interference (EMI), and resistance to corrosion, fiber-optic sensors are becoming more and more desirable when designed with characteristics of assembly-free and operation in the reflection configuration. Additionally, such sensors are also needed in optofluidic/ferrofluidic systems for chemical/biomedical sensing applications. In this chapter, liquid-assisted laser micromachining techniques were investigated for the fabrication of assembly-free, all-optical fiber sensor probes. All-in-fiber optofluidic sensor and fiber in-line ferrofluidic sensor were presented as examples with respect to these laser processing techniques.

Introduction The invention of the ruby laser in 1960 (Maiman 1960) provided much higher laser intensities than previous candidates. Since then, laser has been developed rapidly and used for controllable material processing by high laser intensities. With the advances of mode-locking techniques (Spence et al. 1991) and chirped pulse amplification (Strickland and Mourou 1985), the intensities of commercially available femtosecond (fs) laser systems can be achieved of more than 1013 W/cm2 (Perry and Mourou 1994). At such intensities, any materials, especially for transparent materials, will be ionized and exhibit nonlinear behavior, causing dielectric breakdown and structural change in transparent materials (Itoh et al. 2006). Fs laser is also known as ultrafast laser, due to its unique advantages of ultrashort pulse width (1015 W/cm2 ). Recently, fs laser micromachining has opened up a new avenue for material processing, especially for the transparent materials with large material bandgaps (i.e., fused silica (Eg D 9 eV)). When used for fabrication, fs laser can be used either to remove materials from the surface (ablation) or to modify the properties inside the materials (modification or irradiation). Compared with long pulse lasers (pulses longer than a few picoseconds) (Chichkov et al. 1996), fs laser has many unique characteristics, such as negligible cracks, minimal heat-affected zone, low recast, and high precision.

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Initial studies of fs laser micromachining were first demonstrated in 1987, when ultrafast excimer UV lasers were used to ablate the surface of polymethyl methacrylate (PMMA) (Du et al. 1994). Later, micrometer-sized features on silica (Pronko et al. 1995) and silver surfaces (Chimmalgi et al. 2003) were performed using infrared fs laser systems. In less than 10 years, the resolution of surface ablation has improved to enable high precision with nanometer scale (Küper and Stuke 1987). For the material modification, Hirao’s group firstly demonstrated fs laser processed in the bulk of transparent glass in 1996, and the material modification happened beneath the sample surface, forming waveguiding structures with a permanent refractive index change localized to the focal volume (Davis et al. 1996). This was followed by introducing two-photon polymerization into a resin (Kawata et al. 2001) and printing complex three-dimensional (3D) structures with nanometer-scale resolution. Over the past decade, fs laser micromachining has been used in a broad range of applications, from waveguide writing, cell ablation to biological sample modification (Gattass and Mazur 2008). As a result, fs laser was proven to be a unique and versatile contactless material modification tool. The fiber-optic field has undergone tremendous growth and advancement over the past 50 years. Initially considered as a medium of transmitting light and imagery for medical endoscopic applications, optical fibers were later promoted as an information carrier for telecommunication applications in the mid-1960s (Bates 2001). C. Kao and G. Hockham of the British company Standard Telephones and Cables (STC) were the first to propose the idea that the attenuation in optical fibers could be controlled down to 20 dB/km, allowing fibers to be a good candidate for telecommunication applications (Hecht 2004). Since then, the research and development of optical fiber telecommunication applications have been widened immensely (Udd 1995). Today, optical telecommunication has proven to be the preferred method to transmit vast amounts of data and information with high speed and long-haul capability. In addition to communication applications, optical fibers have been widely used for broad sensing applications in the fields of physical, chemical, and biological analyses, including structure health monitoring, harsh environment temperature sensing, biological and chemical refractive index/pH sensors, and medical imaging (Haque et al. 2014; Liu et al. 2013; Wei et al. 2008; Zhang et al. 2013; Zhou et al. 2010). Optical fiber, most of the time, is made of fused silica glass. It consists of fiber core, fiber cladding, and the outside buffer/jacket layer for protection, as shown in Fig. 1. Due to the small amount of doping elements (i.e., germaniumdoped) in the fiber core area, the refractive index of fiber core (n1 ) is slightly larger than the index of fiber cladding (n2 ), so the light can propagate inside the fiber core due to the so-called total internal reflection (TIR). Compared with electrical sensors, fiber-optic sensors offer many intrinsic advantages, such as small size/lightweight, immunity to electromagnetic interference (EMI), resistance to chemical corrosion, high temperature capability, high sensitivity, and multiplexing and distributed sensing. Fiber-optic sensors have been long envisioned as a cost-effective and reliable candidate for many applications. However, although the past half century has seen a

2354 Fig. 1 Schematic representation of the optical fiber as a waveguide under TIR conditions

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TIR

Core (n1)

Buffer/jacket

Cladding (n2)

steady growth of appearance of optical fibers in various industry sectors, the market of fiber-optic sensors has not blew out as expected. On the contrary, many of the unique advantages and special requirements of advanced fiber-optic sensors and devices are yet to be fully harvested, including assembly-free, inexpensive, and miniaturized fiber-pigtailed sensor probes as well as long-haul, fully distributed fiber-optic sensor networks. Special requirements for advanced fiber-optic sensors are highly needed in biomedical/chemical applications, i.e., the hot topic of optofluidic systems. Aiming to synergistically combine integrated optics and microfluidics, optofluidic-based systems have attracted much research interest because of their unique advantages toward biological/chemical sensing applications (Fan and White 2011). In an optofluidic system, the liquid of interest is constrained and manipulated in a small geometry to interact with the optics. As such, the physical, chemical, and biological properties of the liquid can be probed and analyzed effectively using optical means (Psaltis et al. 2006). Most optofluidic systems have been constructed on a planar platform with microchannels in silica/polymetric materials and probed by a variety of optical means, such as absorbance, fluorescence, refractometry, Raman scattering, etc. (Monat et al. 2007). Objective lenses are commonly used to couple light into and out of the microfluidics (Woolley and Mathies 1994). However, the need of using a microscope to perform the optical alignment limits its field applicability. A number of efforts have been made to fabricate optical waveguides inside the substrate to confine and transport light in the substrate (Osellame et al. 2007) or directly integrate optical fibers with the fluidics for excitation and probing (Domachuk et al. 2006). However, the transmission efficiency of light coupling is still a challenge in most optofluidic configurations. In addition to planar configurations, it has been suggested that the microfluidics can be directly fabricated on an optical fiber to form the so-called all-in-fiber optofluidics. The all-in-fiber configuration has the unique advantage of alignmentfree optics and improved robustness. Examples include the photonic crystal fibers (PCFs) filled with functional fluids in their cladding air voids (Jensen et al.

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2004), the capillary-based optofluidic ring resonator (OFRR) with a microchannel for sample delivery (Zhu et al. 2008), and a miniaturized microchannel directly fabricated on a conventional optical fiber for light-fluid interaction (Lai et al. 2006). While the PCF-based and OFRR-based optofluidic sensors utilize the evanescent fields to probe the fluid, the microchannel on a conventional fiber configuration allows a direct light passage through the liquid. However, PCFs are still expensive, and huge transmission loss will be generated when fusion splicing PCFs with single-mode fibers (SMFs). OFRR structure needs to assemble with an ultrathin and fragile fiber-optic taper to read out the signal, limiting the application within the lab condition. Although the sensor with a microchannel in an SMF can probe the liquid, the sensing mechanism is still based on the intensity modulation, whose sensitivity and detection limit are much smaller and higher, respectively, than those of phase modulation-based sensors. Similar to optofluidic systems, filling the embedded microchannels with functional liquids (i.e., ferrofluid) can also allow people to test surrounding magnetic field. Typically, varying magnetic field can change the properties of ferrofluid (i.e., the permittivity of the liquid) and then make the integrated device function as a ferrofluidic sensor for various applications. Additionally, it’s highly desired that the ferrofluidic sensor can be directly implemented in an all-fiber form with minimum insertion loss and desired performance. In this chapter, first of all, a brief overview of the background physics describing fs laser micromachining of transparent materials was presented. The discussion of laser/matter interactions was starting from atomic scale-free electron plasma formation to energy deposition and material modification. Secondly, the development of a typical home-integrated fs laser micromachining system and preliminary fabrication results were presented in details. After that, methods of liquid-assisted laser processing were illustrated. Buried microfluidic microchannels with sub-microresolution in an SMF can be fabricated either using laser-induced water breakdown technique directly or utilizing laser irradiation followed by chemical etching technique, resulting in fiber sensors for chemical/biomedical sensing application. Finally, an all-in-fiber optofluidic sensor and a fiber in-line ferrofluidic sensor are presented as examples of this technology.

Femtosecond Laser and Material Interactions Free Electron Plasma Formation The most popular femtosecond laser nowadays is Ti:sapphire laser which has its operating wavelength at around 800 nm. High-intensity focused fs laser pulses with the wavelength () of 800 nm have insufficient photon energy (E D h D ¯! D 1.55eV, where ! D 2) to be linearly absorbed in fused silica with the bandgap of material Eg, which is about 9 eV. As such, nonlinear photoionization is dominant to promote electrons from valence band to the conduction band and then generate free electrons. Typically, there are two classes of such

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ionization process: nonlinear photoionization and avalanche ionization (Beresna et al. 2014). Nonlinear photoionization refers to direct electron excitation by an electric laser field. Such process can generate seed electrons that will participate in the next process. Photoionization involves multiphoton ionization and/or tunneling ionization depending on the laser frequency and intensity (Stuart et al. 1995). Once the free electrons exist in the conduction band, free carrier absorption happens, leading to collisional ionization followed by avalanche ionization. In the multiphoton absorption regime, photoionization rate dne /dt strongly depends on the laser intensity: d ne .t; r; z/ D ım .I .t; r; z//m dt

(1)

where t is the time; r, the distance to the Gaussian beam axis; z, the depth from the surface of the bulk material; ne is free electron density; I(t, r, z) is the laser intensity inside the bulk material; m is the order of the multiphoton process, i.e., the number of photons should be six in our case to satisfy the condition E D m¯!  Eg ; and ı m is the cross section of the m-photon absorption. The tunnelling rate, on the other hand, scales more weakly with the laser intensity than the multiphoton rate (Schaffer et al. 2001). The strong dependence on the intensity also means that the photoionization process is more efficient for the laser with short pulse duration. For the long pulse laser (i.e., ns), the photoionization process cannot efficiently generate sufficient seed electrons, and the excitation process becomes strongly reliant on the low concentration of impurities with energy levels which are distributed randomly close to the conduction band. As such, the modification process becomes less deterministic, and precise machining is impossible for longer pulses (Beresna et al. 2014). Avalanche ionization involves free carrier absorption followed by impact ionization. The free electrons in the conduction band oscillate in the electromagnetic field of the laser and gradually gain energy by collisions. After the conduction band electron’s energy exceeds the minimum energy of the conduction band by more than the bandgap energy of the material, it can impact ionize another bound electron from the valence band via collision, resulting in two excited electrons near the bottom of the conduction band (Yablonovitch and Bloembergen 1972). Both electrons can undergo free carrier absorption, impact ionization, and repeat the described energy transfer cycle. Such process will not stop until the laser electric field is dissipated and not strong enough, leading to an electronic avalanche. The density of free electrons generated through the avalanche ionization is: d ne .t; r; z/ D ai I .t; r; z/ ne .t; r; z/ dt

(2)

where ai is the avalanche ionization coefficient (Stuart et al. 1996). The original laser beam before it interacts with the material is assumed to be a Gaussian distribution

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in time and space. Obviously, at z D 0, it is assumed that the laser focus point is at the surface of the material. The optical properties of the highly ionized dielectrics under an fs pulse can be well determined by plasma properties (Rethfeld et al. 2002). Generally, the plasma frequency ! p is defined by (Mao et al. 2004): s !p .ne / D

e 2 ne .t / "0 me

(3)

where ne is also denoted as free carrier density. When the free electron density excited by photoionization approaches a high density (i.e., ! p  !  1021 cm3 ), a large fraction of the remaining fs laser pulse can be absorbed (Schaffer et al. 2001). Assumed 800 nm laser irradiation, the plasma frequency ! p equals the laser frequency ! when the free carrier density ne is approximately 1.7  1021 cm3 , which is also known as the critical density of free electrons (nc (t) D ! 2 "0 me /e2 ). Such critical value can be also used as a criterion in the model of laser-material interaction. It should be noticed that avalanche ionization requires the presence of seed electrons, which can trigger the process and is more efficient with longer pulse durations. The following rate equation has been widely used to describe this nonlinear photoionization process (the combined action of multiphoton excitation and avalanche ionization) (Stuart et al. 1996): d ne .t; r; z/ D ım .I .t; r; z//m C ai I .t; r; z/ ne .t; r; z/ dt

(4)

However, free carrier losses (i.e., self-trapping and recombination) may occur on a time scale comparable to pulse duration (